Vector Variational Principles for Set-Valued Functions

Abstract

Deriving existence results and necessary conditions for approximate solutions of nonlinear optimization problems under week assumptions is an interesting and modern field in optimization theory. It is of interest to show corresponding results for optimization problems without any convexity and compactness assumptions. Ekeland’s variational principle is a very deep assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighborhood of an approximate solution of the original problem. The importance of Ekeland’s variational principle in nonlinear analysis is well known. Especially, this assertion is very useful for deriving necessary conditions under certain differentiability assumptions. In optimal control Ekeland’s principle can be used in order to prove an ε-maximum principle in the sense of Pontryagin and in approximation theory for deriving ε-Kolmogorov conditions.

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11.1 Introduction

Deriving existence results and necessary conditions for approximate solutions of nonlinear optimization problems under week assumptions is an interesting and modern field in optimization theory. It is of interest to show corresponding results for optimization problems without any convexity and compactness assumptions. Ekeland’s variational principle is a very deep assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighborhood of an approximate solution of the original problem. The importance of Ekeland’s variational principle in nonlinear analysis is well known. Especially, this assertion is very useful for deriving necessary conditions under certain differentiability assumptions. In optimal control Ekeland’s principle can be used in order to prove an ε-maximum principle in the sense of Pontryagin and in approximation theory for deriving ε-Kolmogorov conditions.

Below we recall a versatile variant.

Proposition 11.1 (Ekeland’s Variational Principle [21,  22]). 

Let (X,d) be a complete metric space and $f : X \rightarrow \mathbb{R} \cup \{ +\infty \}$ a proper, lower semicontinuous function bounded below. Consider ε > 0 and x ₀ ∈ X such that f(x ₀ ) ≤inf f + ε. Then for every λ > 0 there exists $\overline{x} \in domf$ such that

$$f(\overline{x}) + {\lambda }^{-1}\epsilon d(\overline{x},{x}_{ 0}) \leq f({x}_{0}),\qquad d(\overline{x},{x}_{0}) \leq \lambda ,$$

(11.1)

and

$$f(\overline{x}) < f(x) + {\lambda }^{-1}\epsilon d(\overline{x},x)\quad \forall \,x \in X \setminus \{\overline{x}\}.$$

(11.2)

This means that for λ, ε > 0 and x ₀an ε-approximate solution of the minimization problem

$$f(x) \rightarrow \min \ \textrm{ s.t.}\ x \in X,$$

(11.3)

there exists a new point $\overline{x}$that is not worse than x ₀and belongs to a λ-neighborhood of x ₀, and especially, $\overline{x}$satisfies the variational inequality (11.2). Relation (11.2) says, in fact, that $\overline{x}$minimizes globally $f + {\lambda }^{-1}\epsilon d(\overline{x},\cdot )$, which is nothing else than a Lipschitz perturbation of f(for “smooth” principles, see [11]). Note that $\lambda = \sqrt{\epsilon }$gives a useful compromise in Proposition 11.1. For applications see Sect. 11.5and, e.g., [24,  25,  58,  61,  62].

There are several statements that are equivalent to Ekeland’s variational principle (EVP); see, e.g., [1,  2,  5,  12,  13,  14,  15,  16,  27,  29,  30,  31,  33,  34,  38,  52,  53,  54].

Phelps [54] introduced for ε > 0 the following closed convex cone ${K}_{\epsilon }$in $X \times \mathbb{R}$, where Xis a Banach space:

$${K}_{\epsilon } :=\{ (x,r) \in X \times \mathbb{R}\;\vert \;\epsilon \vert \vert x\vert \vert \leq -r\}$$

(11.4)

(see Fig. 11.1). Sometimes the cone ${K}_{\epsilon }$is called a Phelps cone. Phelps has shown the existence of minimal points of a set $\mathcal{A}\subseteq X \times \mathbb{R}$with respect to ${K}_{\epsilon }$under a closedness assumption (H) and a boundedness assumption (B) concerning $\mathcal{A}$.

Proposition 11.2 (Phelps Minimal-Point Theorem [53,  54]). 

Let X be a Banach space and $\mathcal{A}(\neq \varnothing ) \subseteq X \times \mathbb{R}$ . Assume

• (H) $\mathcal{A}$ is closed

• (B) $\inf \{r \in \mathbb{R}\;\vert \;(x,r) \in \mathcal{A}\} = 0$

Fig. 11.1

Minimal point $(\overline{x},\overline{y})$of a set Awith respect to ${K}_{\epsilon }$

Suppose ε > 0. Then, for any point $({x}_{0},{r}_{0}) \in \mathcal{A}$ there exists a point $(\overline{x},\overline{r}) \in \mathcal{A}$ such that:

• (a) $(\overline{x},\overline{r}) \in \mathcal{A}\cap (({x}_{0},{r}_{0}) + {K}_{\epsilon })$

• (b) $\{(\overline{x},\overline{r})\} = \mathcal{A}\cap ((\overline{x},\overline{r}) + {K}_{\epsilon })$

Remark 11.1.

The assertion (a) in Proposition 11.2can be considered as a domination propertyand assertion (b) describes a minimal point $(\overline{x},\overline{r})$of $\mathcal{A}$with respect to ${K}_{\epsilon }$.

In Phelps [53] and [54] it is shown that Ekeland’s variational principle (Proposition 11.1) is a conclusion of a minimal-point theorem (Proposition 11.2) setting $\mathcal{A} = epif$in Proposition 11.2. We will present extensions of Phelps minimal-point theorem to general product spaces and corresponding variational principles. The aim of this chapter is to give an overview on existing minimal-point theorems and variational principles of Ekeland’s type for set-valued and vector-valued objective functions. In order to show such assertions a main tool is the application of a certain scalarization technique. In the following section we will discuss scalarizing functionals and their properties.

11.2 Preliminaries

Let us recall some notions and notation for sets and functions defined on locally convex spaces. So let (X, τ) be a locally convex space and A ⊆ X. By clA(or ${cl}_{\tau }A$or $\overline{A}$or ${\overline{A}}^{\tau }$), φAand intAand bdAwe denote the closure (with respect to τ when we want to emphasize the topology), the interior and the boundary of A; moreover convAis the convex hull of Aand $\overline{conv}A := cl(conv A)$. As usual, for A, B ⊆ X, a ∈ X, $\Gamma \subseteq \mathbb{R}$and $\alpha \in \mathbb{R}$we set

$$\begin{array}{rcl} & A + B :=\{ a + b\mid a \in A,\ b \in B\},\quad a + B :=\{ a\} + B, & \\ & \Gamma A :=\{ \gamma a\mid \gamma \in \Gamma ,\ a \in A\},\quad \Gamma a := \Gamma \{a\},\quad \alpha A :=\{ \alpha \}A,\quad - A := (-1)A.& \\ \end{array}$$

The recession cone of the nonempty set A ⊆ Xis the set

$${A}_{\infty } :=\{ u \in X\mid x + tu \in A\ \forall x \in A,\ \forall t \in {\mathbb{R}}_{+}\}.$$

It follows easily that ${A}_{\infty }$is a convex cone; ${A}_{\infty }$is also closed when Ais closed. If Ais a closed convex set then ${A}_{\infty } = {\cap }_{t\in \mathbb{P}}t(A - a)$, where $\mathbb{P} := ]0,+\infty [$and a ∈ A(${A}_{\infty }$does not depend on a ∈ A). Moreover, the indicator function associated to the set A ⊆ Xis the function ${\iota }_{A} : X \rightarrow \overline{\mathbb{R}} := \mathbb{R} \cup \{-\infty ,\infty \}$defined by ι _A (x) : = 0 for x ∈ Aand ι _A (x) : = ∞for x ∈ X ∖ A, where $\infty := +\infty$. A cone K ⊆ Xis called pointed if $K \cap (-K) =\{ \mathbf{0}\}$.

Let $f : X \rightarrow \overline{\mathbb{R}}$; the domain and the epigraph of fare defined by

$$domf :=\{ x \in X\mid f(x) < +\infty \},\quad epif :=\{ (x,t) \in X \times \mathbb{R}\mid f(x) \leq t\}.$$

The function fis said to be convexif epifis a convex set, and fis said to be properif domf≠∅and fdoes not take the value − ∞. Of course, fis lower semicontinuousif epifis closed. The class of lower semi-continuous (lsc for short) proper convex functions on Xwill be denoted by Γ(X). Let B ⊆ X; $f : X \rightarrow \overline{\mathbb{R}}$is called B-monotoneif ${x}_{2} - {x}_{1} \in B \Rightarrow \varphi ({x}_{1}) \leq \varphi ({x}_{2})$. Furthermore, fis called strictly B-monotoneif ${x}_{2} - {x}_{1} \in B \setminus \{\mathbf{0}\} \Rightarrow \varphi ({x}_{1}) < \varphi ({x}_{2})$.

We consider a proper closed convex cone K ⊆ Yand k ⁰ ∈ K ∖ ( − K). As usual, we denote

$$\begin{array}{rcl} & & {K}^{+} :=\{ {y}^{{_\ast}}\in {Y }^{{_\ast}}\mid {y}^{{_\ast}}(k) \geq 0\ \forall k \in K\}, \\ & & {K}^{\#} :=\{ {y}^{{_\ast}}\in {Y }^{{_\ast}}\mid {y}^{{_\ast}}(k) >0\ \forall k \in K \setminus \{\mathbf{0}\}\} \\ \end{array}$$

the positive dual cone of the convex cone K ⊆ Yand the quasi interior of K ⁺, respectively.

In Sect. 11.3we show several properties of scalarizing functionals. Motivated by papers on the field of economics, especially production theory (cf. Luenberger [50]) we assume that the sets Aand Kverify the free-disposal condition $A - K = A$included in assumption (A1) introduced in Sect. 11.3.2; for Lipschitz properties of ${\varphi }_{A,{k}^{0}}$(see (11.5) for its definition) we need the strong free-disposal condition $A - (K \setminus \{ 0\}) = intA$, which is a part of assumption (A2). The main results concerning Lipschitz properties are given in Sect. 11.3.4under assumption (A1): First, without convexity assumptions for the closed set A ⊆ Ywe prove that ${\varphi }_{A,{k}^{0}}$is Lipschitz on Yunder the (stronger) assumption k ⁰ ∈  intK(Theorem 11.4); then, assuming that Ais a convex set with nonempty interior and ${k}^{0}\notin {A}_{\infty }$we show that ${\varphi }_{A,{k}^{0}}$is locally Lipschitz on $int(dom{\varphi }_{A}) = \mathbb{R}{k}^{0} + intA$(Proposition 11.5). Moreover, without assuming the convexity of Aand without the assumption k ⁰ ∈  intKwe give a characterization of Lipschitz continuity of ${\varphi }_{A,{k}^{0}}$on a neighbourhood of y ₀ ∈ Yusing the notion of epi-Lipschitz set introduced by Rockafellar [55] (Theorem 11.5). In Sect. 11.3.5we provide formulas for the conjugate and the subdifferential of ${\varphi }_{A,{k}^{0}}$when Ais convex. Using the properties of the scalarizing functionals we present in Sect. 11.4minimal-point theorems and corresponding variational principles. As an application of the Lipschitz properties of ${\varphi }_{A,{k}^{0}}$, we establish necessary conditions for properly efficient solutions of a vector optimization problem in terms of the Mordukhovich subdifferential in Sect. 11.5.2. Taking into account the fact that the conditions in the definition of properly efficient elements are related to the strong free disposal condition in (A2) we get in Theorem 11.15useful properties for the scalarizing functional ${\varphi }_{A,{k}^{0}}$as well as for the Mordukhovich subdifferential of the scalarized objective function.

11.3 Nonlinear Scalarization Functions

In order to show minimal-point theorems and corresponding variational principles in Sect. 11.4we use a scalarization method by means of certain nonlinear functionals. In this section we discuss useful properties of these functionals (cf. Göpfert et al. [32] and Tammer and Zălinescu [63]).

11.3.1 Construction of Scalarizing Functionals

Having a nonempty subset Aof a real linear space Yand an element k ⁰≠0of Y, Gerstewitz (Tammer) and Iwanow [28] introduced the function (see Fig. 11.2)

$${\varphi }_{A} := {\varphi }_{A,{k}^{0}} : Y \rightarrow \overline{\mathbb{R}},\quad {\varphi }_{A,{k}^{0}}(y) :=\inf \{ t \in \mathbb{R}\mid y \in t{k}^{0} + A\},$$

(11.5)

where, as usual, inf∅: = ∞(and $\sup \varnothing := -\infty$); we use also the convention $(+\infty ) + (-\infty ) := +\infty$.

Fig. 11.2

Level sets of the function ${\varphi }_{A,{k}^{0}}$from (11.5), where $A = -K = -{\mathbb{R}}_{+}^{2}$and k ⁰ ∈ int Kholds

This function was used by Chr. Tammer and her collaborators, as well as by D.T. Luc etc., mainly for scalarization of vector optimization problems. Luenberger [50, Definition 4.1] considered

$$\sigma (g;y) :=\inf \{ \xi \in \mathbb{R}\mid y - \xi g \in \mathcal{Y}\},$$

the corresponding function being called the shortage function associated to the production possibility set $\mathcal{Y}\subseteq {\mathbb{R}}^{m}$and $g \in {\mathbb{R}}_{+}^{m} \setminus \{\mathbf{0}\}$. The case when g = (1, …, 1) was introduced earlier by Bonnisseau and Cornet [10]. A similar function is introduced in [50, Definition 2.1] under the name of benefit function.

More recently such a function was considered in the context of mathematical finance beginning with Artzner et. al. [3]; see Heyde [42] and Hamel [39] for more historical facts. Under the name of topical function such functions were studied by Singer and his collaborators (see [59]). We discuss many important properties of ${\varphi }_{A,{k}^{0}}$in Sect. 11.3.2. Moreover, we study local continuity properties in Sect. 11.3.4. Very recently Bonnisseau and Crettez [4] obtained local Lipschitz properties for ${\varphi }_{A,{k}^{0}}$(called Luenberger shortage function in [4]) in a very special case, more general results are given by Tammer and Zălinescu [63]. Of course, ${\varphi }_{A,{k}^{0}}$is a continuous sublinear functional if Ais a proper closed convex cone and k ⁰ ∈  intA(cf. Corollary 11.2) and so ${\varphi }_{A,{k}^{0}}$is Lipschitz continuous. Such Lipschitz properties of ${\varphi }_{A,{k}^{0}}$are of interest also in the case when A ⊆ Yis an arbitrary (convex) set and the interior of the usual ordering cone in Yis empty like in mathematical finance where the acceptance sets are in function spaces as L _p and the corresponding risk measures are formulated by means of ${\varphi }_{A,{k}^{0}}$(see e.g. Föllmer and Schied [26]).

11.3.2 Properties of Scalarization Functions

Throughout this section Yis a separated locally convex space and Y ^∗is its topological dual, K ⊆ Yis a proper closed convex cone, k ⁰ ∈ K ∖ ( − K) and A ⊆ Yis a nonempty set. The cone Kdetermines the order ≤  _K on Ydefined by y ₁ ≤  _K y ₂if y ₂ − y ₁ ∈ K.

Furthermore, we assume that Asatisfies the following condition (see also [4]):

• (A1) Ais closed, satisfies the free-disposal assumption $A - K = A$, and A ≠ Y.

We shall use also the (stronger) condition:

• (A2) Ais closed, satisfies the strong free-disposal assumption $A - (K \setminus \{\mathbf{0}\}) = intA$, and A ≠ Y.

Because $A - K = A \cup (A - (K \setminus \{\mathbf{0}\}))$, we have that $(A2) \Rightarrow (A1)$ .Moreover, the condition $A - (K \setminus \{\mathbf{0}\}) = intA$is equivalent to A − (K ∖ {0}) ⊆  intA.

Remark 11.2.

Assume that the nonempty set Asatisfies assumption (A2). Then Kis pointed, that is, $K \cap (-K) =\{ \mathbf{0}\}$, and $A - \mathbb{P}{k}^{0} \subseteq intA$for k ⁰ ∈ K ∖ {0}.

The last assertion is obvious. For the first one, assume that k ∈ K ∩ ( − K) ∖ {0}. Take a ∈  bdA( ⊆ A); such an aexists because A≠Y. Then ${a}^{{\prime}} := a - k \in intA \subseteq A$, and so $a = {a}^{{\prime}}- (-k) \in intA$, a contradiction.

Remark 11.3.

When Asatisfies condition (A1) or (A2) with respect to Kand k ⁰ ∈ K ∖ ( − K) then Asatisfies condition (A1) or (A2), respectively, with respect to ${\mathbb{R}}_{+}{k}^{0}$. In fact in many situations it is sufficient to take $K = {\mathbb{R}}_{+}{k}^{0}$for some k ⁰ ∈ Y ∖ {0}. In such a situation (A1) [respectively (A2)] means that Ais a closed proper subset of Yand $A - {\mathbb{R}}_{+}{k}^{0} = A$[respectively $A - \mathbb{P}{k}^{0} \subseteq intA$].

The free-disposal condition $A = A - K$shows that $K \subseteq -{A}_{\infty }$. As observed above ${A}_{\infty }$is also closed because Ais closed. Hence $-{A}_{\infty }$is the largest closed convex cone Kverifying the free-disposal assumption $A = A - K$.

The aim of this section is to find a suitable functional $\varphi : Y \rightarrow \mathbb{R}$and conditions such that two given nonempty subsets Aand Hof Ycan be separated by φ.

To A ⊆ Ysatisfying (A1) and k ⁰ ∈ K ∖ ( − K) we associate the function ${\varphi }_{A,{k}^{0}}$defined in (11.5). We consider the set

$${A}^{{\prime}} :=\{ (y,t) \in Y \times \mathbb{R}\mid y \in t{k}^{0} + A\}.$$

The assumption on Ashows that A ^′is of epigraph type,i.e. if (y, t) ∈ A ^′and t ^′ ≥ t, then (y, t ^′) ∈ A ^′. Indeed, if y ∈ tk ⁰ + Aand t ^′ ≥ t, since

$$t{k}^{0} + A = {t}^{{\prime}}{k}^{0} + A - ({t}^{{\prime}}- t){k}^{0} \subseteq {t}^{{\prime}}{k}^{0} + A,$$

(because of (A1)) we obtain that (y, t ^′) ∈ A ^′. Also observe that ${A}^{{\prime}} = {T}^{-1}(A)$, where $T : Y \times \mathbb{R} \rightarrow Y$is the continuous linear operator defined by $T(y,t) := t{k}^{0} + y$. So, if Ais closed (convex, cone), then A ^′is closed (convex, cone). Obviously, the domain of φ _A is the set $\mathbb{R}{k}^{0} + A$and A ^′ ⊆  epiφ _A  ⊆  clA ^′(because A′is of epigraph type), from which it follows that A ^′ =  epiφ _A if Ais closed, and so φ _A is a lower semicontinuous function.

In the next results we collect several useful properties of φ _A (compare Göpfert et al. [32]).

Theorem 11.1.

Assume that K ⊆ Y is a proper closed convex cone, k ⁰ ∈ K ∖ (−K) and A ⊆ Y is a nonempty set. Furthermore, suppose

• (A1) A is closed, satisfies the free-disposal assumption $A - K = A$ , and A≠Y.

Then φ _A (defined in ( 11.5)) is lsc,$dom{\varphi }_{A} = \mathbb{R}{k}^{0} + A$ ,

$$\{y \in Y \mid {\varphi }_{A}(y) \leq \lambda \} = \lambda {k}^{0} + A\quad \forall \,\lambda \in \mathbb{R},$$

(11.6)

and

$${\varphi }_{A}(y + \lambda {k}^{0}) = {\varphi }_{ A}(y) + \lambda \quad \forall \,y \in Y,\ \forall \,\lambda \in \mathbb{R}.$$

(11.7)

Moreover,

• (a) φ _A is convex if and only if A is convex; φ _A (λy) = λφ _A (y) for all λ > 0 and y ∈ Y if and only if A is a cone.

• (b) φ _A is proper if and only if A does not contain lines parallel to k ⁰ , i.e.,

  $$\forall \,y \in Y,\ \exists \,t \in \mathbb{R}\ :\ y + t{k}^{0}\notin A.$$

  (11.8)

• (c) φ _A is finite-valued if and only if A does not contain lines parallel to k ⁰ and

  $$\mathbb{R}{k}^{0} + A = Y.$$

  (11.9)

• (d) Let B ⊆ Y ; φ _A is B-monotone if and only if A − B ⊆ A.

• (e) φ _A is subadditive if and only if A + A ⊆ A.

Proof.

We have already observed that $dom{\varphi }_{A} = \mathbb{R}{k}^{0} + A$and φ _A is lsc when Ais closed. From the definition of φ _A the inclusion ⊇ in (11.6) is obvious, while the converse inclusion is immediate, taking into account the closedness of A. Formula (11.7) follows easily from (11.6).

• (a) Since the operator Tdefined above is onto and $epi{\varphi }_{A} = {T}^{-1}(A)$, we have that epiφ _A is convex (cone) if and only if A = T(epiφ _A ) is so. The conclusion follows.

• (b) We have

  $${\varphi }_{A}(y) = -\infty \Leftrightarrow y \in t{k}^{0} + A\ \forall t \in \mathbb{R} \Leftrightarrow \{ y + t{k}^{0}\mid t \in \mathbb{R}\} \subseteq A.$$

  The conclusion follows.

• (c) The conclusion follows from (b) and the fact that $dom{\varphi }_{A} = \mathbb{R}{k}^{0} + A$.

• (d) Suppose first that A − B ⊆ Aand take y ₁, y ₂ ∈ Ywith y ₂ − y ₁ ∈ B. Let $t \in \mathbb{R}$be such that y ₂ ∈ tk ⁰ + A. Then ${y}_{1} \in {y}_{2} - B \subseteq t{k}^{0} + (A - B) \subseteq t{k}^{0} + A$, and so φ _A (y ₁) ≤ t. Hence φ _A (y ₁) ≤ φ _A (y ₂). Assume now that φ _A is B-monotone and take y ∈ Aand b ∈ B. From (11.6) we have that φ _A (y) ≤ 0. Since $y - (y - b) \in B$, we obtain that φ _A (y − b) ≤ φ _A (y) ≤ 0, and so, using again (11.6), we obtain that y − b ∈ A.

• (e) Suppose first that A + A ⊆ Aand take y ₁, y ₂ ∈ Y. Let ${t}_{i} \in \mathbb{R}$be such that y _i  ∈ t _i k ⁰ + Afor i ∈ { 1, 2}. Then ${y}_{1} + {y}_{2} \in ({t}_{1} + {t}_{2}){k}^{0} + (A + A) \subseteq ({t}_{1} + {t}_{2}){k}^{0} + A$, and so ${\varphi }_{A}({y}_{1} + {y}_{2}) \leq {t}_{1} + {t}_{2}$. It follows that ${\varphi }_{A}({y}_{1} + {y}_{2}) \leq {\varphi }_{A}({y}_{1}) + {\varphi }_{A}({y}_{2})$. Assume now that φ _A is subadditive and take y ₁, y ₂ ∈ A. From (11.6) we have that φ _A (y ₁), φ _A (y ₂) ≤ 0. Since φ _A is subadditive, we obtain that ${\varphi }_{A}({y}_{1} + {y}_{2}) \leq {\varphi }_{A}({y}_{1}) + {\varphi }_{A}({y}_{2}) \leq 0$, and so, using again (11.6), we obtain that y ₁ + y ₂ ∈ A.

Remark 11.4.

From Theorem 11.1we get under assumption (A1) that φ _A is lower semicontinuous,

$$A =\{ y \in Y \mid {\varphi }_{A}(y) \leq 0\},\quad intA \subseteq \{ y \in Y \mid {\varphi }_{A}(y) < 0\},$$

(11.10)

and so

$$bdA = A \setminus intA \supseteq \{ y \in Y \mid {\varphi }_{A}(y) = 0\}.$$

(11.11)

In general the inclusion in (11.11) is strict.

Example 11.1.

Consider $K := {\mathbb{R}}_{+}^{2}$, k ⁰: = (1, 0) and

$$A := \left (] -\infty ,0] \times ] -\infty ,0]\right ) \cup \left ([0,\infty [ \times ] -\infty ,-1]\right ).$$

Then ${\varphi }_{A}(u,v) = -\infty$for v ≤ − 1, φ _A (u, v) = ufor v ∈ ( − 1, 0] and φ _A (u, v) = ∞for v > 0. In particular, ${\varphi }_{A}(0,-1) = -\infty$and (0, − 1) ∈  bdA(see Fig. 11.3).

Theorem 11.2.

Assume that K ⊆ Y is a proper closed convex cone, k ⁰ ∈ K ∖ (−K) and A ⊆ Y is a nonempty set. Furthermore, suppose

• (A2) A is closed, satisfies the strong free-disposal assumption $A - (K \setminus \{\mathbf{0}\}) = intA$ , and A≠Y.

Fig. 11.3

$\overline{y} \in bd\ A$with ${\varphi }_{A}(\overline{y}) = -\infty$in Example 11.1

Then (a), (b), (c) from Theorem 11.1 holds, and moreover

• (f) φ _A is continuous and

  $$\begin{array}{rcl} \{y \in Y \mid {\varphi }_{A}(y) < \lambda \}& =& \lambda {k}^{0} + intA,\quad \forall \,\lambda \in \mathbb{R},\end{array}$$

  (11.12)
  $$\begin{array}{rcl}\{y \in Y \mid {\varphi }_{A}(y) = \lambda \}& =& \lambda {k}^{0} + bdA,\quad \forall \,\lambda \in \mathbb{R}. \end{array}$$

  (11.13)

• (g) If φ _A is proper, then

  $${\varphi }_{A}\text{ is $B$-monotone} \Leftrightarrow A - B \subseteq A \Leftrightarrow bdA - B \subseteq A.$$

  Moreover, if φ _A is finite-valued, then

  $${\varphi }_{A}\text{ strictly $B$-monotone} \Leftrightarrow A - (B \setminus \{\mathbf{0}\}) \subseteq intA \Leftrightarrow bdA - (B \setminus \{\mathbf{0}\}) \subseteq intA.$$

• (h) Assume that φ _A is proper; then

  $${\varphi }_{A}\text{ is subadditive} \Leftrightarrow A + A \subseteq A \Leftrightarrow bdA + bdA \subseteq A.$$

Proof.

Suppose now that (A2) holds.

• (f) Let $\lambda \in \mathbb{R}$and take y ∈ λk ⁰ +  intA. Since y − λk ⁰ ∈  intA, there exists ε > 0 such that $y - \lambda {k}^{0} + \epsilon {k}^{0} \in A$. Therefore φ _A (y) ≤ λ − ε < λ, which shows that the inclusion ⊇ always holds in (11.12). Let $\lambda \in \mathbb{R}$and y ∈ Ybe such that φ _A (y) < λ. There exists $t \in \mathbb{R}$, t < λ, such that y ∈ tk ⁰ + A. It follows with (A2) that $y \in \lambda {k}^{0} + A - (\lambda - t){k}^{0} \subseteq \lambda {k}^{0} + intA$. Therefore (11.12) holds, and so φ _A is upper semicontinuous. Because φ _A is also lower semicontinuous, we have that φ _A is continuous. From (11.6) and (11.12) we obtain immediately that (11.13) holds.

• (g) Let us prove the second part, the first one being similar to that of (and partially proved in) (d). So, let φ _A be finite-valued.

  Assume that φ _A is strictly B-monotone and take y ∈ Aand b ∈ − B ∖ {0}. From (11.6) we have that φ _A (y) ≤ 0, and so, by hypothesis, φ _A (y − b) < 0. Using (11.12) we obtain that y − b ∈  intA. Assume now that bdA − (B ∖ {0}) ⊆  intA. Consider y ₁, y ₂ ∈ Ywith y ₂ − y ₁ ∈ B ∖ {0}. From (11.13) we have that y ₂ ∈ φ _A (y ₂)k ⁰ +  bdA, and so ${y}_{1} \in {\varphi }_{A}({y}_{2}){k}^{0} - (bdA + (B \setminus \{\mathbf{0}\})) \subseteq {\varphi }_{A}({y}_{2}){k}^{0} + intA$. From (11.12) we obtain that φ _A (y ₁) < φ _A (y ₂). The remaining implication is obvious.

• (h) Let φ _A be proper. One has to prove bdA +  bdA ⊆ A $\Rightarrow$φ _A is subadditive. Consider y ₁, y ₂ ∈ Y. If {y ₁, y ₂} ⊄ dom}φ _A , there is nothing to prove; hence let y ₁, y ₂ ∈  dom}φ _A . Then, by (11.13), y _i  ∈ φ _A (y _i )k ⁰ +  bdAfor i ∈ { 1, 2}, and so ${y}_{1} + {y}_{2} \in ({\varphi }_{A}({y}_{1}) + {\varphi }_{A}({y}_{2})){k}^{0} + (bdA + bdA) \subseteq ({\varphi }_{A}({y}_{1}) + {\varphi }_{A}({y}_{2})){k}^{0} + A$. Therefore ${\varphi }_{A}({y}_{1} + {y}_{2}) \leq {\varphi }_{A}({y}_{1}) + {\varphi }_{A}({y}_{2})$.

When k ⁰ ∈  int}Kwe get an additional important property of φ _A (see also Theorem 11.4).

Corollary 11.1.

Assume that K ⊆ Y is a proper closed convex cone, k ⁰ ∈ intK and A ⊆ Y satisfies condition (A1). Then φ _A is finite-valued and continuous.

Proof.

Because k ⁰ ∈  int}Kwe have that $\mathbb{R}{k}^{0} + K = Y$. From Theorem 11.1 (c) it follows that

$$dom{\varphi }_{A} = A + \mathbb{R}{k}^{0} = A - K + \mathbb{R}{k}^{0} = A + Y = Y.$$

Assuming that φ _A is not proper, from Theorem 11.1 (c) we get $y + \mathbb{R}{k}^{0} \subseteq A$for some y ∈ Y. Then $Y = y + \mathbb{R}{k}^{0} - K \subseteq A - K = A$, a contradiction. Hence φ _A is finite-valued.

Moreover, we have that $A - \mathbb{P}{k}^{0} \subseteq A - intK \subseteq int(A - K) = intA$. Applying Theorem 11.2 (f) for Kreplaced by ${\mathbb{R}}_{+}{k}^{0}$we obtain that φ _A is continuous.

From the preceding results we get the following particular case.

Corollary 11.2.

Let K ⊆ Y be a proper closed convex cone and k ⁰ ∈−intK. Then

$${\varphi }_{K} : Y \rightarrow \mathbb{R},\quad \quad {\varphi }_{K}(y) :=\inf \{ t \in \mathbb{R}\mid y \in t{k}^{0} + K\}$$

is a well-defined continuous sublinear function such that for every $\lambda \in \mathbb{R}$ ,

$$\{y \in Y \mid {\varphi }_{K}(y) \leq \lambda \} = \lambda {k}^{0} + K,\quad \{y \in Y \mid {\varphi }_{ K}(y) < \lambda \} = \lambda {k}^{0} + intK.$$

Moreover, φ _K is strictly (−intK)-monotone.

Proof.

The assertions follow using Theorem 11.2and Corollary 11.1applied for A: = Kand Kreplaced by − K. For the last part note that $K + intK = intK$.

Now all preliminaries are done, and we can prove the following nonconvex separation theorem.

Theorem 11.3 (Non-convex Separation Theorem). 

Let A ⊆ Y be a closed proper set with nonempty interior, H ⊆ Y a nonempty set such that H ∩ intA = ∅. Let K ⊆ Y be a proper closed convex cone and k ⁰ ∈ intK. Furthermore, assume

• (A2) A is closed, satisfies the strong free-disposal assumption $A - (K \setminus \{\mathbf{0}\}) = intA$ , and A≠Y.

Then φ _A defined by ( 11.5) is a finite-valued continuous function such that

$${\varphi }_{A}(x) \geq 0 >{\varphi }_{A}(y)\quad \forall \,x \in H,\ \forall \,y \in intA;$$

(11.14)

moreover, φ _A (x) > 0 for every x ∈ intH.

Proof.

By Corollary 11.1φ _A is a finite-valued continuous function. By Theorem 11.2 (f) we have that φA = { y ∈ Y∣φ _A (y) < 0}, and so (11.14) obviously holds.

Take y ∈  int}H; then there exists t > 0 such that y − tk ⁰ ∈ H. From (11.7) and (11.12) we obtain that $0 \leq {\varphi }_{A}(y - t{k}^{0}) = {\varphi }_{A}(y) - t$, whence φ _A (y) > 0.

Of course, if we impose additional conditions on A, we have additional properties of the separating functional φ _A (see Theorems 11.1and 11.2).

11.3.3 Continuity Properties

If Ais a proper closed subset of Y(hence ∅≠A≠Y) and $A - \mathbb{P}{k}^{0} \subseteq intA$, applying Theorem 11.2for $K := {\mathbb{R}}_{+}{k}^{0}$we obtain that φ _A is continuous (on Y) and (11.13) holds. In the next result we characterize the continuity of φ _A at a point y ₀ ∈ Y(compare Tammer and Zălinescu [63]).

Proposition 11.3.

Assume that K ⊆ Y is a proper closed convex cone, k ⁰ ∈ K ∖ (−K) and A ⊆ Y is a nonempty set satisfying condition (A1). Then the function φ _A is (upper semi-) continuous at y ₀ ∈ Y if and only if y ₀ − ]φ _A (y ₀ ),∞[ ⋅ k ⁰ ⊆ intA.

Proof.

If φ _A (y ₀) = ∞it is clear that φ _A is upper semicontinuous at y ₀and the inclusion holds. So let φ _A (y ₀) < ∞.

Assume first that φ _A is upper semicontinuous at y ₀. Let $\lambda \in ]{\varphi }_{A}({y}_{0}),\infty [$. Then there exists a neighbourhood Vof y ₀such that φ _A (y) < λ for every y ∈ V. It follows that for y ∈ Vwe have y ∈ λk ⁰ + A, that is, V ⊆ λk ⁰ + A. Hence y ₀ ∈ λk ⁰ +  int}A, whence y ₀ − λk ⁰ ∈  int}A.

Assume now that y ₀ − ]φ _A (y ₀), ∞[ ⋅k ⁰ ⊆  int}Aand take φ _A (y) < λ < ∞. Then, by our hypothesis, $V := \lambda {k}^{0} + A$is a neighbourhood of y ₀and from the definition of φ _A we have that φ _A (y) ≤ λ for every y ∈ V. Hence φ _A is upper semicontinuous at y ₀.

Corollary 11.3.

Under the hypotheses of Proposition 11.3 assume that φ _A is continuous at y ₀ ∈ bdA. Then φ _A (y ₀ ) = 0.

Proof.

Of course, φ _A (y ₀) ≤ 0. If φ _A (y ₀) < 0, from the preceding proposition we obtain the contradiction ${y}_{0} = {y}_{0} - 0{k}^{0} \in intA$.

11.3.4 Lipschitz Properties

The primary goal of this section is to study local Lipschitz properties of the functional ${\varphi }_{A,{k}^{0}}$under as weak as possible assumptions concerning the subset A ⊆ Yand k ⁰ ∈ Y(compare Tammer and Zălinescu [63]).

When Ais a convex set, as noticed above, φ _A is convex. In such a situation from the continuity of φ _A at a point in the interior of its domain one obtains the local Lipschitz continuity of φ _A on the interior of its domain (if the function is proper). Moreover, when $A = -K$and k ⁰ ∈  int}Kthen (it is well known that) φ _A is a continuous sublinear function, and so φ _A is Lipschitz continuous.

Recently in the case $Y = {\mathbb{R}}^{m}$and for $K = {\mathbb{R}}_{+}^{m}$Bonnisseau–Crettez [4] obtained the Lipschitz continuity of φ _A around a point y ∈  bdAwhen − k ⁰is in the interior of the Clarke tangent cone of Aat y. The (global) Lipschitz continuity of φ _A can be related to a result of Gorokhovik–Gorokhovik [35] established in normed vector spaces as we shall see in the sequel.

Theorem 11.4.

Assume that K ⊆ Y is a proper closed convex cone, k ⁰ ∈ K ∖ (−K) and A ⊆ Y is a nonempty set satisfying condition (A1).

• (a) One has

  $$\begin{array}{rcl}{ \varphi }_{A}(y) \leq {\varphi }_{A}({y}^{{\prime}}) + {\varphi }_{ -K}(y - {y}^{{\prime}})\quad \forall y,{y}^{{\prime}}\in Y.& & \end{array}$$

  (11.15)

• (b) If k ⁰ ∈ intK then φ _A is finite-valued and Lipschitz on Y.

Proof.

• (a) By Theorem 11.1(applied for Aand $A := -K$, respectively) we have that φ _A and φ_− K are lower semicontinuous functions, φ_− K being sublinear and proper.

  Let y, y ^′ ∈ Y. If ${\varphi }_{A}({y}^{{\prime}}) = +\infty$or ${\varphi }_{-K}(y - {y}^{{\prime}}) = +\infty$it is nothing to prove. In the contrary case let t, s ∈ ℝbe such that $y - {y}^{{\prime}}\in t{k}^{0} - K$and y ^′ ∈ sk ⁰ + A. Then, taking into account assumption (A1)

  $$\begin{array}{rcl} y \in t{k}^{0} - K + s{k}^{0} + A = (t + s){k}^{0} + (A - K) = (t + s){k}^{0} + A.& & \\ \end{array}$$

  It follows that φ _A (y) ≤ t + s. Passing to infimum with respect to tand ssatisfying the preceding relations we get (11.15).

• (b) Assume that k ⁰ ∈  int}K. Let V ⊆ Ybe a symmetric closed and convex neighbourhood of 0 such that k ⁰ + V ⊆ Kand let ${p}_{V } : Y \rightarrow \mathbb{R}$be the Minkowski functional associated to V; then p _V is a continuous seminorm and V = { y ∈ Y∣p _V (y) ≤ 1}. Let y ∈ Yand t > 0 such that y ∈ tV. Then ${t}^{-1}y \in V \subseteq {k}^{0} - K$, whence y ∈ tk ⁰ − K. Hence φ_− K (y) ≤ t. Therefore, φ_− K (y) ≤ p _V (y). This inequality confirms that $(\mathbb{R}{k}^{0} - K =)$ $dom{\varphi }_{-K} = Y$. Moreover, since φ_− K is sublinear we get ${\varphi }_{-K}(y) \leq {\varphi }_{-K}({y}^{{\prime}}) + {p}_{V }(y - {y}^{{\prime}})$and so

  $$\left \vert {\varphi }_{-K}(y) - {\varphi }_{-K}({y}^{{\prime}})\right \vert \leq {p}_{ V }(y - {y}^{{\prime}})\quad \forall y,{y}^{{\prime}}\in Y,$$

  (11.16)

  that is, φ_− K is Lipschitz.

  By Corollary 11.1we have that φ _A is finite-valued (and continuous). From (11.15) we have that ${\varphi }_{A}(y) - {\varphi }_{A}({y}^{{\prime}}) \leq {\varphi }_{-K}(y - {y}^{{\prime}}) \leq {p}_{V }(y - {y}^{{\prime}})$, whence (interchanging yand y ^′)

  $$\left \vert {\varphi }_{A}(y) - {\varphi }_{A}({y}^{{\prime}})\right \vert \leq {p}_{ V }(y - {y}^{{\prime}})\quad \forall y,{y}^{{\prime}}\in Y.$$

  (11.17)

  Hence φ _A is Lipschitz continuous (on Y).

Note that the condition A − (K ∖ {0}) ⊆  int}Adoes not imply that φ _A is proper.

Example 11.2.

Take $A :=\{ (x,y) \in {\mathbb{R}}^{2}\mid y \geq -{\left \vert x\right \vert }^{-1}\}$, with the convention ${0}^{-1} := \infty$, and $K := {\mathbb{R}}_{+}{k}^{0}$with ${k}^{0} := (0,-1)$. Then $A - (K \setminus \{ 0\}) = intA$and ${\varphi }_{A}(0,1) = -\infty.$

Note that, with our notation, [4, Proposition 7] asserts that ${\varphi }_{A,{k}^{0}}$is finite and locally Lipschitz provided $Y = {\mathbb{R}}^{n}$, $K = {\mathbb{R}}_{+}^{n}$and k ⁰ ∈  int}K, which is much less than the conclusion of Theorem 11.4(ii).

Of course, in the conditions of Theorem 11.4(ii) we have that $-{k}^{0} \in int{A}_{\infty }$because K ⊆ − A _∞ . In fact we have also a converse of Theorem 11.4(ii).

Proposition 11.4.

Assume that K ⊆ Y is a proper closed convex cone, k ⁰ ∈ K ∖ (−K) and A ⊆ Y is a nonempty set satisfying condition (A1). If φ _A is finite-valued and Lipschitz then − k ⁰ ∈ intA _∞.

Proof.

By hypothesis there exists a closed convex and symmetric neighbourhood Vof 0such that (11.17) holds. We have that A = { y ∈ Y∣φ _A (y) ≤ 0}. Let y ∈ A, v ∈ Vand α ≥ 0. Then

$${\varphi }_{A}(y + \alpha (v - {k}^{0})) \leq {\varphi }_{ A}(y + \alpha v) - \alpha \leq {\varphi }_{A}(y) + \alpha {p}_{V }(v) - \alpha \leq 0$$

because V = { y ∈ Y∣p _V (y) ≤ 1}. Hence V − k ⁰ ⊆ A _∞ , which shows that − k ⁰ ∈  int}A _∞ .

Corollary 11.4.

Under the assumptions of Proposition 11.4, the function φ _A is finite-valued and Lipschitz if and only if − k ⁰ ∈ intA _∞ .

Proof.

The necessity is given by Proposition 11.4. Assume that − k ⁰ ∈  int}A _∞ . Taking $K := -{A}_{\infty }$, using Theorem 11.4(b) we obtain that φ _A is finite-valued and Lipschitz.

If φK≠∅and k ⁰∉φK, φ_− K is not finite-valued, and so it is not Lipschitz. One may ask if the restriction of φ_− K at its domain is Lipschitz. The next examples show that both situations are possible.

Example 11.3.

Take $K = {\mathbb{R}}_{+}^{2}$and k ⁰ = (1, 0). We have that ${\varphi }_{-K}({y}_{1},{y}_{2}) = {y}_{1}$for y ₂ ≤ 0, ${\varphi }_{-K}({y}_{1},{y}_{2}) = \infty$for y ₂ > 0, and so ${\varphi }_{-K}{\vert }_{dom {\varphi }_{-K}}$is Lipschitz.

Example 11.4.

Take $K := \left \{(u,v,w) \in {\mathbb{R}}^{3}\mid v,w \geq 0,\ {u}^{2} \leq vw\right \}$and k ⁰: = (0, 0, 1); then

$${\varphi }_{-K}(x,y,z) = \left \{\begin{array}{lll} \infty &\text{ if}&y >0\text{ or }[y = 0\text{ and }x\neq 0], \\ z &\text{ if}&x = y = 0, \\ z - {x}^{2}/y&\text{ if}&y < 0. \end{array} \right.$$

It is clear that the restriction of φ_− K at its domain is not continuous at (0, 0, 0) ∈  dom}φ_− K and the restriction of φ_− K at the interior of its domain is not Lipschitz. However, φ_− K is locally Lipschitz on the interior of its domain.

The last property mentioned in the previous example is a general one for φ _A when Ais convex.

Proposition 11.5.

Let A be a proper closed subset of Y and k ⁰ ∈ Y ∖{ 0} be such that $A - {\mathbb{R}}_{+}{k}^{0} = A$. If A is convex, has nonempty interior, and does not contain any line parallel with k⁰(or equivalently k⁰∉A_∞), then φ_Ais locally Lipschitz on $int(dom{\varphi }_{A}) = \mathbb{R}{k}^{0} + intA$.

Proof.

Because Adoes not contain any line parallel with k ⁰, φ _A is proper (see Theorem 11.1taking into account assumption (A1)). We know that $dom{\varphi }_{A} = \mathbb{R}{k}^{0} + A$, and so $int(dom{\varphi }_{A}) = int(\mathbb{R}{k}^{0} + A) = \mathbb{R}{k}^{0} + intA$(see, e.g., [67, Exercise 1.4]). On the other hand it is clear that A ⊆ { y ∈ Y∣φ _A (y) ≤ 0}. Since φA≠∅, we have that φ _A is bounded above on a neighbourhood of a point, and so φ _A is locally Lipschitz on $int(dom{\varphi }_{A}) = \mathbb{R}{k}^{0} + intA$(see e.g. [67, Corollary 2.2.13]).

We have seen in Theorem 11.4that φ _A is Lipschitz even if Ais not convex when k ⁰ ∈  int}K. So, in the sequel we are interested by the case in which Ais not convex, k ⁰∉φKand Adoes not contain any line parallel with k ⁰.

Note that for Anot convex and y ∈  int}(dom}φ _A ) we can have situations in which φ _A is not continuous at yor φ _A is continuous but not Lipschitz around y.

Example 11.5.

Take $K := {\mathbb{R}}_{+}^{2}$,k ⁰: = (1, 0) and

$$\begin{array}{rcl}{ A}_{1}& :=& \left (] -\infty ,0] \times ] -\infty ,1]\right ) \cup \left ([0,1] \times ] -\infty ,0]\right ) \\ {A}_{2}& :=& \{(a,b)\mid a \in ]0,\infty [,\ b \leq -{a}^{2}\} \cup \left (] -\infty ,0] \times ] -\infty ,1]\right ).\end{array}$$

Then

$${\varphi }_{{A}_{1},{k}^{0}}(u,v) = \left \{\begin{array}{lll} \infty &\text{ if}&v >1, \\ u &\text{ if}&0 < v \leq 1, \\ u - 1&\text{ if}&v \leq 0, \end{array} \right.\quad {\varphi }_{{A}_{2},{k}^{0}}(u,v) = \left \{\begin{array}{lll} \infty &\text{ if}&v >1, \\ u &\text{ if}&0 < v \leq 1, \\ u -\sqrt{-v}&\text{ if}&v \leq 0. \end{array} \right.$$

It is clear that $(0,0) \in int(dom{\varphi }_{{A}_{1}})$but ${\varphi }_{{A}_{1}}$is not continuous at (0, 0), and $(0,0) \in int(dom{\varphi }_{{A}_{2}})$, ${\varphi }_{{A}_{2}}$is continuous at (0, 0) but ${\varphi }_{{A}_{2}}$is not Lipschitz at (0, 0). 

In what concerns the Lipschitz continuity of φ _A around a point y ∈  dom}φ _A in finite dimensional spaces this can be obtained using the notion of epi-lipschitzianity of a set as introduced by Rockafellar [55] (see also [56]). We extend this notion in our context. We say that the set A ⊆ Yis epi-Lipschitz at $\overline{y} \in A$in the direction v ∈ Y ∖ {0} if there exist ε > 0 and a (closed convex symmetric) neighbourhood V ₀of 0in Ysuch that

$$\forall y \in (\overline{y} + {V }_{0}) \cap A,\ \forall w \in v + {V }_{0},\ \forall \lambda \in [0,\epsilon ] :\ y + \lambda w \in A.$$

(11.18)

Note that (11.18) holds for v = 0if and only if $\overline{y} \in intA$. Moreover, if $\overline{y} \in intA$then Ais epi-Lipschitz at $\overline{y} \in A$in any direction.

Theorem 11.5.

Let A be a proper closed subset of Y and k ⁰ ∈ Y ∖{ 0} be such that $A - {\mathbb{R}}_{+}{k}^{0} = A$. Assume that y₀∈ Y is such that φ_A(y₀) ∈ ℝ. Then φ_Ais finite and Lipschitz on a neighbourhood of y₀if and only if A is epi-Lipschitz at $\overline{y} := {y}_{0} - {\varphi }_{A}({y}_{0}){k}^{0}$in the direction − k⁰.

Proof.

Using (11.7) we get ${\varphi }_{A}(\overline{y}) = 0$. Recall also that A = { y ∈ Y∣φ _A (y) ≤ 0} and the finite values of φ _A are attained (because Ais closed).

Assume that there exist a closed convex symmetric neighbourhood Vof 0in Yand $p : Y \rightarrow \mathbb{R}$a continuous seminorm such that φ _A is finite on y ₀ + Vand $\left \vert {\varphi }_{A}(y) - {\varphi }_{A}({y}^{{\prime}})\right \vert \leq p(y - {y}^{{\prime}})$for all y, y ^′ ∈ y ₀ + V. Taking into account (11.7), we have that φ _A is finite on $\overline{y} + V$and

$$\left \vert {\varphi }_{A}(y) - {\varphi }_{A}({y}^{{\prime}})\right \vert \leq p(y - {y}^{{\prime}})\quad \forall y,{y}^{{\prime}}\in \overline{y} + V.$$

Take V ₀: = { y ∈ {1} {3}V∣p(y) ≤ 1} and ε ∈ ]0, 1] such that εk ⁰ ∈ V ₀. Let us show that (11.18) holds with vreplaced by − k ⁰. For this take $y \in (\overline{y} + {V }_{0}) \cap A$, $w \in -{k}^{0} + {V }_{0}$and λ ∈ [0, ε]. Then $y - \lambda {k}^{0} -\overline{y} \in {V }_{0} + {V }_{0} \subseteq V$and $y + \lambda w -\overline{y} = y - \lambda {k}^{0} -\overline{y} + \lambda (w + {k}^{0}) \in {V }_{0} + {V }_{0} + {V }_{0} \subseteq V$, and so

$$\begin{array}{rcl}{ \varphi }_{A}(y + \lambda w)& \leq & {\varphi }_{A}(y - \lambda {k}^{0}) + p(\lambda (w + {k}^{0})) = {\varphi }_{ A}(y) - \lambda + \lambda p(w + {k}^{0}) \\ & \leq & \lambda (p(w + {k}^{0}) - 1) \leq 0. \end{array}$$

Hence y + λw ∈ A.

Assume now that (11.18) holds with vreplaced by − k ⁰. Let r ∈ ]0, ε] be such that 2r(1 + p(k ⁰)) < 1, where $p := {p}_{{V }_{0}}$. Of course, {y∣p(y) ≤ λ} = λV ₀for every λ > 0 and if p(y) = 0 then y ∈ λV ₀for every λ > 0. Set

$$M :=\{ y \in \overline{y} + r{V }_{0}\mid \left \vert {\varphi }_{A}(y)\right \vert \leq p(y -\overline{y})\};$$

of course, $\overline{y} \in M$. We claim that $M = \overline{y} + r{V }_{0}$. Consider y ∈ M, w ∈ V ₀and λ ∈ [0, r]. Setting ${y}^{{\prime}} := y - {\varphi }_{A}(y){k}^{0} \in A$, we have that φ _A (y ^′) = 0 and

$$p({y}^{{\prime}}-\overline{y}) \leq p(y -\overline{y}) + \left \vert {\varphi }_{ A}(y)\right \vert \cdot p({k}^{0}) \leq r\left (1 + p({k}^{0})\right ) < \frac{1} {2} \leq 1,$$

(11.19)

and so, by (11.18), ${y}^{{\prime}} + \lambda (w - {k}^{0}) \in A$; hence φ _A (y ^′ + λw) ≤ λ.

Take v ∈ rV ₀. On one hand one has

$${\varphi }_{A}({y}^{{\prime}} + v) = {\varphi }_{ A}\left ({y}^{{\prime}} + p(v) \cdot \frac{1} {p(v)}v\right ) \leq p(v)$$

if p(v) > 0, and ${\varphi }_{A}({y}^{{\prime}} + v) = {\varphi }_{A}({y}^{{\prime}} + \lambda ({\lambda }^{-1}v)) \leq \lambda$for every λ ∈ ]0, r], whence ${\varphi }_{A}({y}^{{\prime}} + v) \leq 0 = p(v)$. Therefore, φ _A (y ^′ + v) ≤ p(v).

On the other hand, assume that ${\varphi }_{A}({y}^{{\prime}} + v) < -p(v)$. Because 2r(1 + p(k ⁰)) < 1, there exists t > 0 such that $r + (t + r)p({k}^{0}) \leq 1/2$and ${\varphi }_{A}({y}^{{\prime}} + v) < -p(v) - t =: {t}^{{\prime}} < 0$. It follows that ${y}^{{\prime}} + v - {t}^{{\prime}}{k}^{0} \in A$. Moreover, taking into account (11.19),

$$p({y}^{{\prime}}+v-{t}^{{\prime}}{k}^{0}-\overline{y}) \leq p({y}^{{\prime}}-\overline{y})+p(v)+(t+p(v))p({k}^{0}) \leq 1/2+r+(t+r)p({k}^{0}) \leq 1,$$

and so ${y}^{{\prime}} + v - {t}^{{\prime}}{k}^{0} \in (\overline{y} + {V }_{0}) \cap A$. Using (11.18), if p(v) > 0 then

$${y}^{{\prime}} + t{k}^{0} = {y}^{{\prime}}-\left ({t}^{{\prime}} + p(v)\right ){k}^{0} = {y}^{{\prime}} + v - {t}^{{\prime}}{k}^{0} + p(v)\left (-{k}^{0} - \frac{1} {p(v)}v\right ) \in A,$$

while if p(v) = 0 then

$${y}^{{\prime}} + (1 - \gamma )t{k}^{0} = {y}^{{\prime}} + v - {t}^{{\prime}}{k}^{0} + \gamma t\left (-{k}^{0} - {(\gamma t)}^{-1}v\right ) \in A$$

for $\gamma :=\min \{ \tfrac{1} {2},\epsilon {t}^{-1}\}$. We get the contradiction $0 = {\varphi }_{A}({y}^{{\prime}}) \leq -t < 0$in the first case and $0 = {\varphi }_{A}({y}^{{\prime}}) \leq -t(1 - \gamma ) < 0$in the second case. Hence φ _A (y ^′ + v) ∈ ℝand $\left \vert {\varphi }_{A}({y}^{{\prime}} + v) - {\varphi }_{A}({y}^{{\prime}})\right \vert \leq p(v)$for every v ∈ rV ₀, or equivalently,

$${\varphi }_{A}(y + v) \in \mathbb{R},\quad \left \vert {\varphi }_{A}(y + v) - {\varphi }_{A}(y)\right \vert \leq p(v)\quad \forall v \in r{V }_{0}.$$

(11.20)

When $y := \overline{y} \in M$, from (11.20) we get $\overline{y} + r{V }_{0} \subseteq M$, and so $M = \overline{y} + r{V }_{0}$as claimed. Moreover, if $y,{y}^{{\prime}}\in \overline{y} + \tfrac{1} {2}r{V }_{0}$, then y ∈ Mand ${y}^{{\prime}} = y + v$for some v ∈ rV ₀; using again (11.20) we have that $\left \vert {\varphi }_{A}({y}^{{\prime}}) - {\varphi }_{A}(y)\right \vert \leq p({y}^{{\prime}}- y)$. The conclusion follows.

The next result is similar to Corollary 11.3.

Corollary 11.5.

Let A be a proper closed subset of Y and k ⁰ ∈ Y ∖{ 0} be such that $A - {\mathbb{R}}_{+}{k}^{0} = A$. Consider $\overline{y} \in bdA$. If A is epi-Lipschitz at $\overline{y}$in the direction − k⁰then ${\varphi }_{A}(\overline{y}) = 0$.

Proof.

Consider ε ∈ ]0, 1[ and V ₀provided by (11.18) with $v := -{k}^{0}$. Assume that ${\varphi }_{A}(\overline{y})\neq 0$. Then there exists t > 0 such that $t{p}_{{V }_{0}}({k}^{0}) \leq \epsilon$and $y := \overline{y} + t{k}^{0} \in A$. Taking λ : = tin (11.18) we obtain that $y + t(-{k}^{0} + {V }_{0}) = \overline{y} + t{V }_{0} \subseteq A$, contradicting the fact that $\overline{y} \in bdA$.

Corollary 11.6.

Let A be a proper closed subset of Y and k ⁰ ∈ Y ∖{ 0} be such that $A - {\mathbb{R}}_{+}{k}^{0} = A$. Assume that dim Y < ∞ and $\overline{y} \in bdA$ . Then φ _A is finite and Lipschitz on a neighbourhood of $\overline{y}$ if and only if $-{k}^{0} \in int{T}_{Cl}(A,\overline{y})$ , where ${T}_{Cl}(A,\overline{y})$ is the Clarke tangent cone of A at $\overline{y}$ .

Proof.

By [56, Theorem 2I], $-{k}^{0} \in int{T}_{Cl}(A,\overline{y})$if and only if Ais epi-Lipschitz at $\overline{y}$in the direction − k ⁰. The conclusion follows from Corollary 11.3, Theorem 11.5and Corollary 11.5.

The fact that φ _A is Lipschitz on a neighbourhood of $\overline{y}$under the condition $-{k}^{0} \in int{T}_{Cl}(A,\overline{y})$is obtained in [4, Proposition 6] in the case $Y = {\mathbb{R}}^{m}$(and $K = {\mathbb{R}}_{+}^{m}$).

Consider y ^∗ ∈ Y ^∗such that $\left \langle {k}^{0},{y}^{{_\ast}}\right \rangle \neq 0$, H: = kery ^∗and take

$${\varphi }_{0} : H \rightarrow \overline{\mathbb{R}},\quad {\varphi }_{0}(z) := {\varphi }_{A}(z),$$

that is, φ₀ = φ _A  |  _H . Since φ _A is lsc, so is φ₀. Then any y ∈ Ycan be written uniquely as z − tk ⁰with z ∈ Hand $t \in \mathbb{R}$. So, by (11.7), ${\varphi }_{A}(y) = {\varphi }_{A}(z - t{k}^{0}) = {\varphi }_{0}(z) - t$. Using (11.10) we obtain that $A =\{ z - t{k}^{0}\mid (z,t) \in epi{\varphi }_{0}\}$. Conversely, if $g : H \rightarrow \overline{\mathbb{R}}$is a lsc function and $A :=\{ z - t{k}^{0}\mid (z,t) \in epig\}$, then Ais a closed set with $A - {\mathbb{R}}_{+}{k}^{0} = A$and φ₀ = g. Therefore, the closed set Awith the property $A - {\mathbb{R}}_{+}{k}^{0} = A$is uniquely determined by a lsc function ${\varphi }_{0} : H \rightarrow \overline{\mathbb{R}}$. Moreover, for $\overline{y} = \overline{z} -\overline{t}{k}^{0}$we have that φ _A is finite (resp. continuous) at $\overline{y}$if and only if φ₀is finite (resp. continuous) at $\overline{z}$. Moreover, because $Y = H + \mathbb{R}{k}^{0}$and the sum is topological (that is, the projection onto Hparallel to ℝk ⁰is continuous), we have that φ _A is finite and Lipschitz continuous on a neighbourhood of $\overline{y}$if and only if φ₀is finite and Lipschitz continuous on a neighbourhood of $\overline{z}$. Similarly, φ _A is finite and Lipschitz continuous if and only if φ₀is finite and Lipschitz continuous.

Note that for Ya normed vector space in [35] one says that Ais (globally) epi-Lipschitz in the direction e ∈ Y ∖ {0} if there exist a closed linear subspace Hof codimension 1 with e ∉ Hand a Lipschitz function $g : H \rightarrow \mathbb{R}$such that $A =\{ y + \alpha e\mid y \in H,\ \alpha \in \mathbb{R},\ g(y) \leq \alpha \}$; Ais epi-Lipschitz if there exists e ∈ Y ∖ {0} such that Ais epi-Lipschitz in the direction e. The main result of [35] asserts that the proper closed set A ⊆ Yis epi-Lipschitz in the direction eif and only if e ∈  int}A _∞ , and so A ⊆ Yis epi-Lipschitz if and only if φA _∞ ≠∅.

The discussion above shows that not only the main theorem of [35] can be obtained from Corollary 11.4, but this one extends the main theorem of [35] to locally convex spaces.

11.3.5 The Formula for the Conjugate and Subdifferential of φ _A  for A Convex

The results of this section (less the second part of Corollary 11.7) were established in several papers; we give the proofs for reader’s convenience. The formula for the conjugate of φ _A is derived by Hamel [40, Theorem 3] and can be related also to [57, Theorem 3] and [60, Theorem 2.2]. Results concerning the subdifferential of φ _A are given in [17, Theorem 2.2, Lemma 2.1]. Another proof of these assertions using the formula for the conjugates is presented in Hamel [40, Corollary 12]. In the statements below we use some usual notation from convex analysis. So, having Xa separated locally convex space with topological dual X ^∗and $f : X \rightarrow \overline{\mathbb{R}}$, the conjugate of fis the function ${f}^{{_\ast}} : {X}^{{_\ast}}\rightarrow \overline{\mathbb{R}}$defined by ${f}^{{_\ast}}({x}^{{_\ast}}) :=\sup \left \{{x}^{{_\ast}}(x) - f(x)\mid x \in X\right \}$and its subdifferential at x ∈ Xwith f(x) ∈ ℝis the set $\partial f(x) :=\{ {x}^{{_\ast}}\in {X}^{{_\ast}}\mid {x}^{{_\ast}}({x}^{{\prime}}- x) \leq f({x}^{{\prime}}) - f(x)\ \forall x\prime \in X\}$; ∂f(x) : = ∅if f(x)∉ℝ. Having a set A ⊆ X, the indicator of Ais the function ${\iota }_{A} : X \rightarrow \overline{\mathbb{R}}$defined by ι _A (x) : = 0 for x ∈ Aand ι _A (x) : = ∞for x ∈ X ∖ A, while the support of Ais the function σ _A : = (ι _A )^∗. When Ais nonempty the domain of σ _A is a convex cone which is called the barrier cone of Aand is denoted by bar}A. Moreover, the normal cone of Aat a ∈ Ais the set N(A, a) : = ∂ι _A (a).

Proposition 11.6.

Let A be a proper closed subset of Y and k ⁰ ∈ Y ∖{ 0} be such that $A - {\mathbb{R}}_{+}{k}^{0} = A$. Assume that A is convex and k⁰∉A_∞. Then φ_A∈ Γ(Y ), that is, φ_Ais a proper lsc convex function,

$${ \varphi }_{A}^{{_\ast}}({y}^{{_\ast}}) = \left \{\begin{array}{ll} {\sigma }_{A}({y}^{{_\ast}})&\text{ if }{y}^{{_\ast}}\in barA,\ {y}^{{_\ast}}({k}^{0}) = 1, \\ \infty &\text{ otherwise,} \end{array} \right.$$

(11.21)

and $\partial {\varphi }_{A}(y) \subseteq \{ {y}^{{_\ast}}\in barA\mid {y}^{{_\ast}}({k}^{0}) = 1\} \subseteq \{ {y}^{{_\ast}}\in {K}^{+}\mid {y}^{{_\ast}}({k}^{0}) = 1\}$for every y ∈ Y.

Proof.

From [32, Theorem 2.3.1]) we have that φ _A  ∈ Γ(Y). Consider y ^∗ ∈ Y ^∗. Then

$$\begin{array}{rcl}{ \varphi }_{A}^{{_\ast}}({y}^{{_\ast}})& =& \sup \left \{{y}^{{_\ast}}(y) - {\varphi }_{ A}(y)\mid y \in Y \right \} \\ & =& \sup \left \{{y}^{{_\ast}}(y) - t\mid y \in Y,\ t \in \mathbb{R},\ y \in t{k}^{0} + A\right \} \\ & =& \sup \left \{{y}^{{_\ast}}(t{k}^{0} + a) - t\mid y \in Y,\ t \in \mathbb{R},\ a \in A\right \} \\ & =& \sup \{{y}^{{_\ast}}(a)\mid a \in A\} +\sup \{ t({y}^{{_\ast}}({k}^{0}) - 1)\mid t \in \mathbb{R}\}.\end{array}$$

Hence (11.21) holds.

Since ∂f(y) ⊆  dom}f ^∗for every proper function $f : Y \rightarrow \overline{\mathbb{R}}$and every y ∈ Y, the first estimate for ∂φ _A (y) follows. Moreover, because $A = A - K$we have ${\sigma }_{A} = {\sigma }_{A-K} = {\sigma }_{A} + {\sigma }_{-K} = {\sigma }_{A} + {\iota }_{{K}^{+}}$, and so bar}A ⊆ K ⁺.

The estimate bar}A ⊆ K ⁺becomes more precise when $K = -{A}_{\infty }$; in fact one has ${({A}_{\infty })}^{+} = -{cl}_{{w}^{{_\ast}}}(barA)$. Indeed, from [67, Exercise 2.23] we have that ${\iota }_{{A}_{\infty }} = {({\iota }_{A})}_{\infty } = {\sigma }_{dom {\iota }_{A}^{{_\ast}}} = {\sigma }_{dom {\sigma }_{A}}$, whence ${\iota }_{-{({A}_{\infty })}^{+}} = {({\iota }_{{A}_{\infty }})}^{{_\ast}} = {\iota }_{{cl }_{{w}^{{_\ast}}}(dom {\sigma }_{A})}$, and so ${cl}_{{w}^{{_\ast}}}(dom{\sigma }_{A}) = -{({A}_{\infty })}^{+}$.

Using Proposition 11.6one deduces the expression of ∂φ _A (see also [17, Theorem 2.2] for Ya normed vector space).

Corollary 11.7.

Assume that A is convex and k ⁰ ∉A _∞ . Then for all $\overline{y} \in Y$ one has

$$\partial {\varphi }_{A}(\overline{y}) =\{ {y}^{{_\ast}}\in barA\mid {y}^{{_\ast}}({k}^{0}) = 1,\ {y}^{{_\ast}}(\overline{y}) - {\varphi }_{ A}(\overline{y}) \geq {y}^{{_\ast}}(y)\ \forall y \in A\}.$$

(11.22)

Moreover, if (A2) holds then ∂φ _A (y) ⊆ K ^# for every y ∈ Y.

Proof.

Fix $\overline{y} \in Y$. If $\overline{y}\notin dom{\varphi }_{A}$then both sets in (11.22) are empty. Let $\overline{y} \in dom{\varphi }_{A}$. Then, of course, $\overline{y} - {\varphi }_{A}(\overline{y}){k}^{0} \in A$. If ${y}^{{_\ast}}\in \partial {\varphi }_{A}(\overline{y})$then ${\varphi }_{A}(\overline{y}) + {\varphi }_{A}^{{_\ast}}({y}^{{_\ast}}) = {y}^{{_\ast}}(\overline{y})$. Taking into account (11.21) we obtain that

$${y}^{{_\ast}}\in barA,\ {y}^{{_\ast}}({k}^{0}) = 1\ \text{ and}\ {y}^{{_\ast}}(\overline{y}) - {\varphi }_{ A}(\overline{y}) \geq {y}^{{_\ast}}(y)\ \forall y \in A,$$

(11.23)

that is, the inclusion ⊆ holds in (11.22). Conversely, if y ^∗ ∈ Y ^∗is such that (11.23) holds, since $\overline{y} - {\varphi }_{A}(\overline{y}){k}^{0} \in A$and y ^∗(k ⁰) = 1, we obtain that ${y}^{{_\ast}}\left (\overline{y} - {\varphi }_{A}(\overline{y}){k}^{0}\right ) = {\sigma }_{A}({y}^{{_\ast}})$, which shows that ${\varphi }_{A}(\overline{y}) + {\varphi }_{A}^{{_\ast}}({y}^{{_\ast}}) = {y}^{{_\ast}}(\overline{y})$. Hence ${y}^{{_\ast}}\in \partial {\varphi }_{A}(\overline{y})$. Therefore, (11.22) holds.

Assume now that (A2) holds, that is, A − (K ∖ { 0}) ⊆  int}A, and take ${y}^{{_\ast}}\in \partial {\varphi }_{A}(\overline{y})$. Hence $\overline{y} \in dom{\varphi }_{A}$. Consider k ∈ K ∖ { 0}. Since $(\overline{y} - k) -\overline{y} = -k \in -(K \setminus \{ 0\})$, by Theorem 11.4 (iv), we have that ${y}^{{_\ast}}(-k) \leq {\varphi }_{A}(\overline{y} - k) - {\varphi }_{A}(\overline{y}) < 0$, that is, y ^∗(k) > 0. Therefore, y ^∗ ∈ K ^#.

11.4 Minimal-Point Theorems and Corresponding Variational Principles

11.4.1 Introduction

The celebrated Ekeland variational principle [21] (see Proposition 11.1) has many equivalent formulations and generalizations.

The aim of this section is to show general minimal-point theorems and corresponding variational principles. In Proposition 11.2an existence result for minimal points of a set $\mathcal{A}$with respect to the cone ${K}_{\epsilon }$defined by (11.4) is presented. Taking into account (11.4) we get

$$({x}_{1} - {x}_{2},{r}_{1} - {r}_{2}) \in {K}_{\epsilon }\quad \Longleftrightarrow\quad \epsilon \vert \vert {x}_{1} - {x}_{2}\vert \vert \leq -({r}_{1} - {r}_{2}).$$

This means

$${r}_{2} \geq {r}_{1} + \epsilon \vert \vert {x}_{1} - {x}_{2}\vert \vert.$$

(11.24)

Quite rapidly after the publication of the Ekeland variational principle (EVP) in 1974 there were formulated extensions to functions $f : (X,d) \rightarrow Y$, where Yis a real (topological) vector space. A systematization of such results was done in [34] (see also [32]), where instead of a function fit was considered a subset of X×Y; said differently, it was considered a multifunction from Xto Y. In [32] we have shown minimal-point theorems in product spaces X×Ywith respect to a relation

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{{k}^{0}}({x}_{2},{y}_{2})\;\Longleftrightarrow\;{y}_{2} \in {y}_{1} + d({x}_{1},{x}_{2}){k}^{0} + K,$$

(11.25)

where Kis the convex ordering cone in Yand k ⁰ ∈ K ∖ {0}. This is an extension of the binary relation defined by (11.24) to product spaces X×Y. Very recently the term d(x ₁, x ₂)k ⁰in (11.25) was replaced by d(x, x ^′)Hwith Ha bounded convex subset of K(see [8]) or by F(x, x ^′) ⊆ K, Fbeing a so called K-metric (see [36]); in both papers one deals with functions $f : X \rightarrow Y$.

In order to formulate general minimal-point theorems in this section we replace d(x ₁, x ₂)k ⁰in (11.25) by a set-valued map Fwith certain properties (compare Tammer and Zălinescu [64]).

It is worth mentioning that a weaker result than a full (= authentic) minimal-point theoremgives an EVP, as shown in this section. Such a weaker result is called not authentic minimal-point theorem.

In this section we present new results with proofs very similar to the corresponding ones in [34], which have as particular cases most part of the existing EVPs, or they are very close to them. Moreover, we use the same approach to get extensions of EVPs of Isac–Tammer and Ha types, as well as extensions of EVPs for bi-functions.

In the sequel (X, d) is a complete metric space, Yis a real topological vector space, Y ^∗is its topological dual, and K ⊆ Yis a proper convex cone.

If Yis just a real linear space we endow it with the finest locally convex topology, that is, the topology defined by all the seminorms on Y. 

As in [6] and [7], we say that E ⊆ Yis quasi bounded(from below) if there exists a bounded set B ⊆ Ysuch that E ⊆ B + K; as in [36], we say that Eis K-bounded(by scalarization) if y ^∗(E) is bounded from below for every y ^∗ ∈ K ⁺. It is clear that any quasi bounded set is K-bounded.

Let $F : X \times X \rightrightarrows K$satisfy the conditions:

• (F1) 0 ∈ F(x, x) for all x ∈ X

• (F2) $F({x}_{1},{x}_{2}) + F({x}_{2},{x}_{3}) \subseteq F({x}_{1},{x}_{3}) + K$for all x ₁, x ₂, x ₃ ∈ X

Using Fwe introduce a preorder on X×Y, denoted by ≼  _F , in the following manner:

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{F}({x}_{2},{y}_{2})\;\Longleftrightarrow\;{y}_{2} \in {y}_{1} + F({x}_{1},{x}_{2}) + K.$$

(11.26)

Indeed, ≼  _F is reflexive by (F1). If (x ₁, y ₁) ≼  _F (x ₂, y ₂) and (x ₂, y ₂) ≼  _F (x ₃, y ₃), then

$${y}_{2} = {y}_{1} + {v}_{1} + {k}_{1},\quad {y}_{3} = {y}_{2} + {v}_{2} + {k}_{2}$$

(11.27)

with v ₁ ∈ F(x ₁, x ₂), v ₂ ∈ F(x ₂, x ₃) and k ₁, k ₂ ∈ K. By (F2) we have that ${v}_{1} + {v}_{2} = {v}_{3} + {k}_{3}$for some v ₃ ∈ F(x ₁, x ₃) and k ₃ ∈ K, and so

$${y}_{3} = {y}_{1} + {v}_{1} + {k}_{1} + {v}_{2} + {k}_{2} = {y}_{1} + {v}_{3} + {k}_{1} + {k}_{2} + {k}_{3} \in {y}_{1} + F({x}_{1},{x}_{3}) + K;$$

hence (x ₁, y ₁) ≼  _F (x ₃, y ₃), and so ≼  _F is transitive. Of course,

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{F}({x}_{2},{y}_{2}) \Rightarrow {y}_{1} {\leq }_{K}{y}_{2};$$

(11.28)

moreover, by (F1), we have that

$$(x,{y}_{1}) {\preccurlyeq }_{F}(x,{y}_{2})\;\Longleftrightarrow\;{y}_{2} \in {y}_{1} + K\;\Longleftrightarrow\;{y}_{1} {\leq }_{K}{y}_{2}.$$

(11.29)

Besides conditions (F1) and (F2) we shall assume to be true the condition

• (F3) There exists z ^∗ ∈ K ⁺such that

  $$\eta (\delta ) :=\inf \left \{{z}^{{_\ast}}(v)\mid v \in {\cup }_{ d(x,{x}^{{\prime}})\geq \delta }F(x,{x}^{{\prime}})\right \} >0\quad \forall \delta >0.$$

  (11.30)

Clearly, by (F3) we have that 0∉cl} conv}F(x, x ^′) for x≠x ^′. 

A sufficient condition for (11.30) is

$${ \inf }_{z\in F(x,{x}^{{\prime}})}{z}^{{_\ast}}(z) \geq d(x,{x}^{{\prime}})\quad \forall x,{x}^{{\prime}}\in X.$$

(11.31)

If (11.31) holds then

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{F}({x}_{2},{y}_{2}) \Rightarrow d({x}_{1},{x}_{2}) \leq {z}^{{_\ast}}({y}_{ 2}) - {z}^{{_\ast}}({y}_{ 1}).$$

(11.32)

Indeed, since F(x ₁, x ₂) ⊆ K, from (11.28) we get first that y ₁ ≤  _K y ₂; then from (11.27)

$${z}^{{_\ast}}({y}_{ 2}) = {z}^{{_\ast}}({y}_{ 1})+{z}^{{_\ast}}({v}_{ 1})+{z}^{{_\ast}}({k}_{ 1}) \geq {z}^{{_\ast}}({y}_{ 1}){+\inf }_{v\in F({x}_{1},{x}_{2})}{z}^{{_\ast}}(v) \geq {z}^{{_\ast}}({y}_{ 1})+d({x}_{1},{x}_{2}),$$

and so (11.32) holds.

Using (11.32) we obtain that

$$\left [({x}_{1},{y}_{1}) {\preccurlyeq }_{F}({x}_{2},{y}_{2}),\ ({x}_{2},{y}_{2}) {\preccurlyeq }_{F}({x}_{1},{y}_{1})\right ] \Rightarrow \left [{x}_{1} = {x}_{2},\ {z}^{{_\ast}}({y}_{ 1}) = {z}^{{_\ast}}({y}_{ 2})\right ];$$

(11.33)

moreover, if z ^∗ ∈ K ^#then ≼  _F is anti-symmetric, and so ≼  _F is a partial order.

For Fsatisfying conditions (F1)–(F3), z ^∗being that from (F3), we introduce the order relation ${\preccurlyeq }_{F,{z}^{{_\ast}}}$on X×Yby

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{F,{z}^{{_\ast}}}({x}_{2},{y}_{2})\;\Longleftrightarrow\;\left \{\begin{array}{l} ({x}_{1},{y}_{1}) = ({x}_{2},{y}_{2})\text{ or} \\ ({x}_{1},{y}_{1}) {\preccurlyeq }_{F}({x}_{2},{y}_{2})\text{ and }{z}^{{_\ast}}({y}_{1}) < {z}^{{_\ast}}({y}_{2}). \end{array} \right.$$

(11.34)

It is easy to verify that ${\preccurlyeq }_{F,{z}^{{_\ast}}}$is reflexive, transitive, and antisymmetric.

11.4.2 Minimal Points in Product Spaces

We take X, Y, K, Fas above, that is, Fsatisfies conditions (F1)–(F3), z ^∗being that from (F3).

Consider a nonempty set $\mathcal{A}\subseteq X \times Y$. In the sequel we shall use the condition (H1) on $\mathcal{A}$, where ℕis the set of nonnegative integers; moreover, we set ${\mathbb{N}}^{{_\ast}} := \mathbb{N} \setminus \{ 0\}$.

The next theorem is the main result of this section.

Theorem 11.6 (Minimal-Point Theorem with Respect to ${\preccurlyeq }_{<Emphasis Type="Bold">\text{ {\it F}}</Emphasis>,{<Emphasis Type="Bold">\text{ {\it z}}</Emphasis>}^{{_\ast}}}$). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper convex cone. Let $F : X \times X \rightrightarrows K$ satisfy conditions (F1)–(F3) and $\mathcal{A}\subseteq X \times K$ satisfy the condition

• (H1) For every ≼ _F -decreasing sequence $(({x}_{n},{y}_{n})) \subseteq \mathcal{A}$ with ${x}_{n} \rightarrow x \in X$ there exists y ∈ Y such that $(x,y) \in \mathcal{A}$ and (x,y) ≼ _F (x _n ,y _n ) for every $n \in \mathbb{N}$.

Furthermore, suppose that

• (B1) z ^∗ (from (F3)) is bounded from below on ${\Pr }_{Y }(\mathcal{A})$.

Then for every $({x}_{0},{y}_{0}) \in \mathcal{A}$ there exists an element $(\overline{x},\overline{y})$ of $\mathcal{A}$ such that:

• (a) $(\overline{x},\overline{y}) {\preccurlyeq }_{F,{z}^{{_\ast}}}({x}_{0},{y}_{0})$

• (b) $(\overline{x},\overline{y})$ is a minimal element of $\mathcal{A}$ with respect to ${\preccurlyeq }_{F,{z}^{{_\ast}}}$

Proof.

Let

$$\alpha :=\inf \left \{{z}^{{_\ast}}(y)\mid \exists x \in X : (x,y) \in \mathcal{A},\ (x,y) {\preccurlyeq }_{ F,{z}^{{_\ast}}}({x}_{0},{y}_{0})\right \} \in \mathbb{R}.$$

Let us denote by $\mathcal{A}(x,y)$the set of those $({x}^{{\prime}},{y}^{{\prime}}) \in \mathcal{A}$with $({x}^{{\prime}},{y}^{{\prime}}) {\preccurlyeq }_{F,{z}^{{_\ast}}}(x,y)$. We construct a sequence ${(({x}_{n},{y}_{n}))}_{n\geq 0} \subseteq \mathcal{A}$as follows: Having $({x}_{n},{y}_{n}) \in \mathcal{A}$, we take $({x}_{n+1},{y}_{n+1}) \in \mathcal{A}({x}_{n},{y}_{n})$such that

$${z}^{{_\ast}}({y}_{ n+1}) \leq \inf \{ {z}^{{_\ast}}(y)\mid (x,y) \in \mathcal{A}({x}_{ n},{y}_{n})\} + 1/(n + 1).$$

Of course, ((x _n , y _n )) is ${\preccurlyeq }_{F,{z}^{{_\ast}}}$-decreasing. It follows that (y _n ) _n ≥ 0is ≤  _K -decreasing, and so the sequence ${\left ({z}^{{_\ast}}({y}_{n})\right )}_{n\geq 0}$is nonincreasing and bounded from below; hence $\gamma :=\lim {z}^{{_\ast}}({y}_{n}) \in \mathbb{R}$.

If $\mathcal{A}({x}_{{n}_{0}},{y}_{{n}_{0}})$is a singleton (that is, $\{({x}_{{n}_{0}},{y}_{{n}_{0}})\}$) for some n ₀ ∈ ℕ, then clearly $(\overline{x},\overline{y}) := ({x}_{{n}_{0}},{y}_{{n}_{0}})$is the desired element. In the contrary case the sequence (z ^∗(y _n )) is (strictly) decreasing; moreover, γ < z ^∗(y _n ) for every n ∈ ℕ.

Assume that (x _n ) is not a Cauchy sequence. Then there exist δ > 0 and the sequences (n _k ), (p _k ) from ${\mathbb{N}}^{{_\ast}}$such that ${n}_{k} \rightarrow \infty$and $d({x}_{{n}_{k}},{x}_{{n}_{k}+{p}_{k}}) \geq \delta$for every k. Since $({x}_{{n}_{k}+{p}_{k}},{y}_{{n}_{k}+{p}_{k}}) {\preccurlyeq }_{F,{z}^{{_\ast}}}({x}_{{n}_{k}},{y}_{{n}_{k}})$we obtain that

$${z}^{{_\ast}}({y}_{{ n}_{k}}) - {z}^{{_\ast}}({y}_{{ n}_{k}+{p}_{k}}) \geq \inf \left \{{z}^{{_\ast}}(v)\mid v \in F({x}_{{ n}_{k}+{p}_{k}},{x}_{{n}_{k}})\right \} \geq \eta (\delta )\quad \forall k \in \mathbb{N}.$$

Since η(δ) > 0 and (z ^∗(y _n )) is convergent, this is a contradiction. Therefore, (x _n ) is a Cauchy sequence in the complete metric space (X, d), and so (x _n ) converges to some $\overline{x} \in X$. Since ((x _n , y _n )) is ≼  _F -decreasing, by (H1) there exists some $\overline{y} \in Y$such that $(\overline{x},\overline{y}) \in \mathcal{A}$and $(\overline{x},\overline{y}) {\preccurlyeq }_{F}({x}_{n},{y}_{n})$for every n ∈ ℕ. It follows that ${z}^{{_\ast}}(\overline{y}) \leq \lim {z}^{{_\ast}}({y}_{n})$, and so ${z}^{{_\ast}}(\overline{y}) < {z}^{{_\ast}}({y}_{n})$for every $n \in \mathbb{N}$. Therefore $(\overline{x},\overline{y}) {\preccurlyeq }_{F,{z}^{{_\ast}}}({x}_{n},{y}_{n})$for every $n \in \mathbb{N}$. Let $({x}^{{\prime}},{y}^{{\prime}}) \in \mathcal{A}$be such that $({x}^{{\prime}},{y}^{{\prime}}) {\preccurlyeq }_{F,{z}^{{_\ast}}}(\overline{x},\overline{y})$. Since $(\overline{x},\overline{y}) {\preccurlyeq }_{F,{z}^{{_\ast}}}({x}_{n},{y}_{n})$, we have that $({x}^{{\prime}},{y}^{{\prime}}) {\preccurlyeq }_{F,{z}^{{_\ast}}}({x}_{n},{y}_{n})$for every $n \in \mathbb{N}$. It follows that

$$0 \leq {z}^{{_\ast}}(\overline{y}) - {z}^{{_\ast}}({y}^{{\prime}}) \leq {z}^{{_\ast}}({y}_{ n}) - {z}^{{_\ast}}({y}^{{\prime}}) \leq 1/n\quad \forall \,n \geq 1,$$

whence ${z}^{{_\ast}}({y}^{{\prime}}) = {z}^{{_\ast}}(\overline{y})$. By the definition of ${\preccurlyeq }_{F,{z}^{{_\ast}}}$we obtain that $({x}^{{\prime}},{y}^{{\prime}}) = (\overline{x},\overline{y})$.

As seen from the proof, for a fixed $({x}_{0},{y}_{0}) \in \mathcal{A}$it is sufficient that z ^∗be bounded from below on the set $\{y \in Y \mid \exists x \in X : (x,y) \in \mathcal{A},\ (x,y) {\preccurlyeq }_{F,{z}^{{_\ast}}}({x}_{0},{y}_{0})\}$instead of being bounded from below on ${\Pr }_{Y }(\mathcal{A}).$

Remark 11.5.

When k ⁰ ∈ Kand $F(x,{x}^{{\prime}}) :=\{ d(x,{x}^{{\prime}}){k}^{0}\}$we have that Fsatisfies conditions (F1) and (F2); moreover, if Yis a separated locally convex space and − k ⁰∉{cl}K, then there exists z ^∗ ∈ K ⁺with z ^∗(k ⁰) = 1, and so (F3) is also satisfied (even (11.31) is satisfied). In this case condition (H1) becomes condition (H1) in [32, p. 199]. So Theorem 11.6extends [32, Theorem 3.10.7] to this framework, using practically the same proof.

In [36] one considers for a proper pointed convex cone D ⊆ Ya so called set-valued D-metric, that is, a multifunction $F : X \times X \rightrightarrows D$satisfying the following conditions:

• (i) F(x, y)≠∅and F(x, x) = { 0} ∀x, y ∈ X, and 0∉F(x, y) ∀x≠y

• (ii) F(x, y) = F(y, x) ∀x, y ∈ X

• (iii) $F(x,y) + F(y,z) \subseteq F(x,z) + D$∀x, y, z ∈ X

The basic supplementary assumptions on Dand Fare:

• (S1) Dis w-normal and D _F is based.

• (S2) $\mathbf{0}\notin {cl}_{w}\left ({\cup }_{d(x,y)\geq \delta }F(x,y)\right )$ $\forall \delta >0.$

Here ${K}_{F} := cone(conv \left (\cup \{F(x,y)\mid x,y \in X\}\right ))$and ${D}_{F} := \left ({K}_{F} \setminus \{\mathbf{0}\} + D\right ) \cup \{\mathbf{0}\}.$

As observed in [36], Dis w-normal iff ${D}^{+} - {D}^{+} = {Y }^{{_\ast}}$, and D _F is based iff D ⁺ ∩ K _F ^#≠∅. 

Comparing with our assumptions on F, we see that (F1) is weaker than (i) because we ask just 0 ∈ F(x, x) for every x ∈ X, and we don’t ask the symmetry condition (ii). From (F3) we obtain that (S2) is verified and that z ^∗ ∈ K _F ^#and so $\left ({K}_{F} \setminus \{\mathbf{0}\} + K\right ) \cup \{\mathbf{0}\}$is based, but we don’t need either Kbe w-normal or even Kbe pointed.

Another possible choice for F, considered also in [36], is F(x, x ^′) : = d(x, x ^′)Hwith H ⊆ K ∖ {0} a nonempty set such that H + Kis convex. Clearly (F1), (i), and (ii) are satisfied (for D = K). From the convexity of H + Kwe obtain easily that (F2) (and (iii)) holds. When Yis a separated locally convex space condition (F3) is equivalent to 0∉cl}(H + K). In order to have that (S1) holds one needs ${K}^{+} - {K}^{+} = {Y }^{{_\ast}}$and the existence of z ^∗ ∈ K ⁺with z ^∗(v) > 0 for every v ∈ H, while for (S2) one needs 0∉cl} _w H(see [36, Lemma 5.9 (d)]); of course, if $H = H + K$, the last condition is equivalent to 0∉cl}(H + K). So it seems that our condition (F3) is more convenient than (S1) and (S2).

For Has above, that is, H ⊆ Kis a nonempty set such that H + Kis convex and 0∉cl}(H + K), we consider F _H (x, x ^′) : = d(x, x ^′)Hfor x, x ^′ ∈ X, and we set

$${\preccurlyeq }_{H} :={\preccurlyeq }_{{F}_{H}};$$

moreover, if z ^∗ ∈ K ⁺is such that infz ^∗(H) > 0 we set

$${\preccurlyeq }_{H,{z}^{{_\ast}}} :={\preccurlyeq }_{{F}_{H},{z}^{{_\ast}}}.$$

An immediate consequence of the preceding theorem is the next result.

Corollary 11.8 (Minimal-Point Theorem with Respect to ${\preccurlyeq }_{<Emphasis Type="Bold">\text{ {\it H}}</Emphasis>,{<Emphasis Type="Bold">\text{ {\it z}}</Emphasis>}^{{_\ast}}}$). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space, K ⊆ Y is a proper convex cone and $\mathcal{A}\subseteq X \times Y$ satisfies:

• (H1) For every ≼ _H -decreasing sequence $(({x}_{n},{y}_{n})) \subseteq \mathcal{A}$ with ${x}_{n} \rightarrow x \in X$ there exists y ∈ Y such that $(x,y) \in \mathcal{A}$ and (x,y) ≼ _H (x _n ,y _n ) for every $n \in \mathbb{N}$.

Suppose that there exists z ^∗ ∈ K ⁺ such thatinf z ^∗ (H) > 0 and

• (B1) $\inf {z}^{{_\ast}}\left ({\Pr }_{Y }(\mathcal{A})\right ) >-\infty$.

Then for every $({x}_{0},{y}_{0}) \in \mathcal{A}$ there exists $(\overline{x},\overline{y}) \in \mathcal{A}$ such that:

• (a) $(\overline{x},\overline{y}) {\preccurlyeq }_{H,{z}^{{_\ast}}}({x}_{0},{y}_{0})$ .

• (b) $(\overline{x},\overline{y})$ is a minimal element of $\mathcal{A}$ with respect to ${\preccurlyeq }_{H,{z}^{{_\ast}}}$.

A condition related to (H1) is the next one.

• (H2) For every sequence $\left (({x}_{n},{y}_{n})\right ) \subseteq \mathcal{A}$with ${x}_{n} \rightarrow x \in X$and (y _n ) ≤  _K -decreasing there exists y ∈ Ysuch that $(x,y) \in \mathcal{A}$and y ≤  _K y _n for every n ∈ ℕ.

Remark 11.6.

Note that (H2) holds if $\mathcal{A}$is closed with ${\Pr }_{Y }(\mathcal{A}) \subseteq {y}_{0} + K$for some y ₀ ∈ Yand every ≤  _K -decreasing sequence in Kis convergent (i.e., Kis a sequentially Daniell cone). In fact, instead of asking that $\mathcal{A}$is closed we may assume that

$$\forall {\left (({x}_{n},{y}_{n})\right )}_{n\geq 1} \subseteq \mathcal{A} : \left [{x}_{n} \rightarrow x,\ {y}_{n} \rightarrow y,\ ({y}_{n})\ \text{ is } {\leq }_{K}\text{ -decreasing } \Rightarrow (x,y) \in \mathcal{A}\right ].$$

Remark 11.7.

Note that (H1) is verified whenever $\mathcal{A}$satisfies (H2) and

$$\forall u \in X,\ \forall X \supseteq ({x}_{n}) \rightarrow x \in X\ :\ { \bigcap \nolimits }_{n\in \mathbb{N}}\left (F({x}_{n},u) + K\right ) \subseteq F(x,u) + K.$$

Indeed, let $\left (({x}_{n},{y}_{n})\right ) \subseteq \mathcal{A}$be ≼  _F -decreasing with ${x}_{n} \rightarrow x$. It is obvious that (y _n ) is ≤  _K -decreasing. By (H2), there exists y ∈ Ysuch that $(x,y) \in \mathcal{A}$and y ≤  _K y _n for every n ∈ ℕ. It follows that

$${y}_{n} \in {y}_{n+p} + F({x}_{n+p},{x}_{n}) + K \subseteq y + F({x}_{n+p},{x}_{n}) + K\quad \forall \,n,p \in \mathbb{N}.$$

Fix n; then ${y}_{n} - y \in F({x}_{n+p},{x}_{n}) + K$for every $p \in \mathbb{N}$, and so, by our hypothesis, ${y}_{n} - y \in F(x,{x}_{n}) + K$because ${\lim }_{p\rightarrow \infty }{x}_{n+p} = x$. Therefore, (x, y) ≼  _F (x _n , y _n ). 

Remark 11.8.

In the case F = F _H , (H1) is verified whenever $\mathcal{A}$satisfies (H2) and H + Kis closed.

Indeed, let $\left (({x}_{n},{y}_{n})\right ) \subseteq \mathcal{A}$be a ≼  _H -decreasing sequence with ${x}_{n} \rightarrow x$. It is obvious that (y _n ) is ≤  _K -decreasing. By (H2), there exists y ∈ Ysuch that $(x,y) \in \mathcal{A}$and y ≤  _K y _n for every $n \in \mathbb{N}$.

Fix n. If x _n  = xthen clearly (x, y) = (x _n , y) ≼  _H (x _n , y _n ). Else, because $d({x}_{n+p},{x}_{n}) \rightarrow d(x,{x}_{n}) >0$for $p \rightarrow \infty$, we get d(x _n + p , x _n ) > 0 for sufficiently large p, and so

$${y}_{n} \in {y}_{n+p}+d({x}_{n+p},{x}_{n})H+K \subseteq y+d({x}_{n+p},{x}_{n})H+K = y+d({x}_{n+p},{x}_{n})(H+K)$$

for sufficiently large p. Since H + Kis closed we obtain that

$${y}_{n} \in y + d({x}_{n},x)(H + K) = y + d({x}_{n},x)H + K,$$

that is, (x, y) ≼  _H (x _n , y _n ). 

Another condition to be added to (H2) in order to have (H1) is suggested by the hypotheses of [8, Theorem 4.1]. Recall that a set C ⊆ Yis cs-complete (see[67, p. 9]) if for all sequences ${({\lambda }_{n})}_{n\geq 1} \subseteq [0,\infty )$and ${({y}_{n})}_{n\geq 1} \subseteq C$such that ${\sum \nolimits }_{n\geq 1}{\lambda }_{n} = 1$and the sequence ${\left ({\sum \nolimits }_{m=1}^{n}{\lambda }_{m}{y}_{m}\right )}_{n\geq 1}$is Cauchy, the series ∑ _n ≥ 1λ _n y _n is convergent and its sum belongs to C. One says that C ⊆ Yis cs-closed if the sum of the series ∑ _n ≥ 1λ _n y _n belongs to Cwhenever ∑ _n ≥ 1λ _n y _n is convergent and (y _n ) ⊆ C, ${({\lambda }_{n})}_{n\geq 1} \subseteq [0,\infty )$and ∑ _n ≥ 1λ _n  = 1. Of course, any cs-complete set is cs-closed; if Yis complete then the converse is true. Moreover, notice that any cs-closed set is convex.

Note that the sequence ${\left ({\sum \nolimits }_{m=1}^{n}{\lambda }_{m}{y}_{m}\right )}_{n\geq 1}$is Cauchy whenever ${({\lambda }_{n})}_{n\geq 1} \subseteq [0,\infty )$is such that the series ∑ _n ≥ 1λ _n is convergent and (y _n ) _n ≥ 1 ⊆ Yis such that conv}{y _n ∣n ≥ 1} is bounded; of course, if Yis a locally convex space then B ⊆ Yis bounded iff conv} Bis bounded. Indeed, let ${({\lambda }_{n})}_{n\geq 1} \subseteq [0,\infty )$with ∑ _n ≥ 1λ _n convergent and (y _n ) _n ≥ 1 ⊆ Ywith B: =  conv}{y _n ∣n ≥ 1} bounded. Fix V ⊆ Ya balanced neighborhood of 0. Because Bis bounded, there exists α > 0 such that B ⊆ αV. Since the series ∑ _n ≥ 1λ _n is convergent there exists n ₀ ≥ 1 such that ${\sum \nolimits }_{k=n}^{n+p}{\lambda }_{k} \leq {\alpha }^{-1}$for all $n,p \in \mathbb{N}$with n ≥ n ₀. Then for such n, pand some b _n, p  ∈ Bwe have

$${\sum \nolimits }_{k=n}^{n+p}{\lambda }_{ k}{y}_{k} = \left ({\sum \nolimits }_{k=n}^{n+p}{\lambda }_{ k}\right ){b}_{n,p} \in [0,{\alpha }^{-1}]B \subseteq [0,{\alpha }^{-1}]\alpha V = V.$$

Proposition 11.7.

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper closed convex cone. Furthermore, suppose that H ⊆ K is a nonempty cs-complete bounded set with 0∉cl(H + K). If $\mathcal{A}$satisfies (H2) then $\mathcal{A}$satisfies (H1), too.

Proof.

Let ${\left (({x}_{n},{y}_{n})\right )}_{n\geq 1} \subseteq \mathcal{A}$be a ≼  _H -decreasing sequence with ${x}_{n} \rightarrow x$. It follows that (y _n ) is ≤  _K -decreasing. By (H2), there exists y ∈ Ysuch that $(x,y) \in \mathcal{A}$and y ≤  _K y _n for every $n \in \mathbb{N}$.

Because ${\left (({x}_{n},{y}_{n})\right )}_{n\geq 1}$is ≼  _H -decreasing we have that

$${y}_{n} = {y}_{n+1} + d({x}_{n},{x}_{n+1}){h}_{n} + {k}_{n}$$

(11.35)

with h _n  ∈ Hand k _n  ∈ Kfor n ≥ 1. If ${x}_{n} = {x}_{\overline{n}}$for $n \geq \overline{n} \geq 1$we take $x := {x}_{\overline{n}}$; then (x, y) ≼  _H (x _n , y _n ) for every $n \in \mathbb{N}$. Indeed, for $n \leq \overline{n}$we have that $({x}_{\overline{n}},{y}_{\overline{n}}) {\preccurlyeq }_{H}({x}_{n},{y}_{n});$because $y {\leq }_{K}{y}_{\overline{n}}$, by (11.29) we get $(x,y) {\preccurlyeq }_{H}(x,{y}_{\overline{n}}) = ({x}_{\overline{n}},{y}_{\overline{n}})$, and so (x, y) ≼ (x _n , y _n ). If $n >\overline{n}$, using again (11.29), we have (x, y) = (x _n , y) ≼  _H (x _n , y _n ). 

Assume that (x _n ) is not constant for large n. Fix n ≥ 1. From (11.35), for p ≥ 0, we have

$$\begin{array}{rcl}{ y}_{n}& =& {y}_{n+p+1} +{ \sum \nolimits }_{l=n}^{n+p}d({x}_{ l},{x}_{l+1}){h}_{l} +{ \sum \nolimits }_{l=n}^{n+p}{k}_{ l} = {y}_{n+p+1} + \left ({\sum \nolimits }_{l=n}^{n+p}d({x}_{ l},{x}_{l+1})\right ){h}_{n,p} +{ \sum \nolimits }_{l=n}^{n+p}{k}_{ l} \\ & =& y + {k}_{n,p}^{{\prime}} + \left ({\sum \nolimits }_{l=n}^{n+p}d({x}_{ l},{x}_{l+1})\right ){h}_{n,p} \end{array}$$

(11.36)

for some h _n, p  ∈ Hand k _n, p ^′ ∈ K. Assuming that ${\sum \nolimits }_{l\geq n}d({x}_{l},{x}_{l+1}) = \infty$, from

$${\left ({\sum \nolimits }_{l=n}^{n+p}d({x}_{ l},{x}_{l+1})\right )}^{-1}({y}_{ n}-y) = {h}_{n,p}+{\left ({\sum \nolimits }_{l=n}^{n+p}d({x}_{ l},{x}_{l+1})\right )}^{-1}{k}_{ n,p}^{{\prime}}\in H+K,$$

we get the contradiction 0 ∈  cl}(H + K) taking the limit for $p \rightarrow \infty$. Hence $0 < \mu :={ \sum \nolimits }_{l\geq n}d({x}_{l},{x}_{l+1}) < \infty$. Set ${\lambda }_{l} := {\mu }^{-1}d({x}_{l},{x}_{l+1})$for l ≥ n. Since His cs-complete and conv}{h _l ∣l ≥ n} ( ⊆ H) is bounded we obtain that the series ∑ _l ≥ n λ _l h _l is convergent and its sum ${\overline{h}}_{n}$belongs to H. It follows that ${\sum \nolimits }_{l\geq n}d({x}_{l},{x}_{l+1}){h}_{l} = \mu {\overline{h}}_{n}$, and so

$${\overline{k}}_{n} :{=\lim }_{p\rightarrow \infty }{k}_{p}^{{\prime}} = {y}_{ n} - y - \mu {\overline{h}}_{n} \in K$$

because Kis closed. Since $d({x}_{n},{x}_{n+p}) \leq {\sum \nolimits }_{l=n}^{n+p-1}d({x}_{l},{x}_{l+1})$, we obtain that d(x _n , x) ≤ μ, and so

$${y}_{n} = y + d({x}_{n},x){\overline{h}}_{n} +{ \overline{k}}_{n} + \left (\mu - d({x}_{n},x)\right ){\overline{h}}_{n} \in y + d({x}_{n},x)H + K.$$

Hence $(x,y) {\preccurlyeq }_{H}({x}_{n},{y}_{n})$for every $n \in \mathbb{N}$.

The most part of vector EVP type results are established for Ya separated locally convex space. However, there are topological vector spaces Ywhose topological dual reduce to {0}. In such a case it is not possible to find z ^∗satisfying the conditions of Corollary 11.8. In [6, Theorem 1], in the case His a singleton, the authors consider such a situation.

Theorem 11.7 (Not Authentic Minimal-Point Theorem with Respect to ${\preccurlyeq }_{<Emphasis Type="Bold">\text{ {\it H}}</Emphasis>}$). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space. Let K ⊆ Y be a proper closed convex cone and H ⊆ K be a nonempty cs-complete bounded set with 0∉cl(H + K). Suppose that $\mathcal{A}\subseteq X \times Y$satisfies

• (H2) For every sequence $\left (({x}_{n},{y}_{n})\right ) \subseteq \mathcal{A}$ with ${x}_{n} \rightarrow x \in X$ and (y _n ) ≤ _K -decreasing there exists y ∈ Y such that $(x,y) \in \mathcal{A}$ and y ≤ _K y _n for every n ∈ ℕ

and

• (B2) ${\Pr }_{Y }(\mathcal{A})$ is quasi bounded.

Then for every $({x}_{0},{y}_{0}) \in \mathcal{A}$ there exists $(\overline{x},\overline{y}) \in \mathcal{A}$ such that:

• (a) $(\overline{x},\overline{y}) {\preccurlyeq }_{H}({x}_{0},{y}_{0})$

• (b) $(x,y) \in \mathcal{A}$ ,$(x,y) {\preccurlyeq }_{H}(\overline{x},\overline{y})$ imply $x = \overline{x}$

Proof.

First observe that $\mathcal{A}$satisfies condition (H1) by Proposition 11.7. Moreover, because ${\Pr }_{Y }(\mathcal{A})$is quasi bounded, there exists a bounded set B ⊆ Ysuch that ${\Pr }_{Y }(\mathcal{A}) \subseteq B + K.$

Note that for $(x,y) \in \mathcal{A}$the set ${\Pr }_{X}(\mathcal{A}(x,y))$is bounded, where $\mathcal{A}(x,y) :=\{ ({x}^{{\prime}},{y}^{{\prime}}) \in \mathcal{A}\mid ({x}^{{\prime}},{y}^{{\prime}}) {\preccurlyeq }_{H}(x,y)\}$. In the contrary case there exists a sequence ${\left (({x}_{n},{y}_{n})\right )}_{n\geq 1} \subseteq \mathcal{A}(x,y)$with $d({x}_{n},x) \rightarrow \infty$. Hence $y = {y}_{n} + d({x}_{n},x){h}_{n} + {k}_{n} = {b}_{n} + d({x}_{n},x){h}_{n} + {k}_{n}^{{\prime}}$with h _n  ∈ H, b _n  ∈ B, k _n , k _n ^′ ∈ K. It follows that $d{({x}_{n},x)}^{-1}(y - {b}_{n}) \in H + K$, whence the contradiction 0 ∈  cl}(H + K). 

Let us construct a sequence ${(({x}_{n},{y}_{n}))}_{n\geq 0} \subseteq \mathcal{A}$in the following way: Having $({x}_{n},{y}_{n}) \in \mathcal{A}$, where $n \in \mathbb{N}$, because ${D}_{n} :{=\Pr }_{X}\left (\mathcal{A}({x}_{n},{y}_{n})\right )$is bounded, there exists $({x}_{n+1},{y}_{n+1}) \in \mathcal{A}({x}_{n},{y}_{n})$such that

$$d({x}_{n+1},{x}_{n}) \geq \frac{1} {2}\sup \{d(x,{x}_{n})\mid x \in {D}_{n}\} \geq \tfrac{1} {4} diam{D}_{n}.$$

We obtain in this way the sequence ${(({x}_{n},{y}_{n}))}_{n\geq 0} \subseteq \mathcal{A}$, which is ≼  _H -decreasing. Since $\mathcal{A}({x}_{n+1},{y}_{n+1}) \subseteq \mathcal{A}({x}_{n},{y}_{n})$, we have that D _n + 1 ⊆ D _n for every $n \in \mathbb{N}$. Of course, x _n  ∈ D _n . Let us show that $diam{D}_{n} \rightarrow 0$. In the contrary case there exists δ > 0 such that diam}D _n  ≥ 4δ, and so d(x _n + 1, x _n ) ≥ δ for every $n \in \mathbb{N}$. As in the proof of Proposition 11.7, for every $p \in \mathbb{N}$, we obtain that

$$\begin{array}{rcl}{ y}_{0}& =& {y}_{p+1} + \left ({\sum \nolimits }_{l=0}^{p}d({x}_{ l},{x}_{l+1})\right ){h}_{p} +{ \sum \nolimits }_{l=0}^{p}{k}_{ l} = {b}_{p} + \left ({\sum \nolimits }_{l=0}^{p}d({x}_{ l},{x}_{l+1})\right ){h}_{p} + {k}_{p}^{{\prime}} \\ & =& {b}_{p} + (p + 1)\delta {h}_{p} + {k}_{p}^{{\prime\prime}}, \\ \end{array}$$

where h _p  ∈ H, b _p  ∈ B, k _l , k _p ^′, k _p ^′′ ∈ K. It follows that ${[(p + 1)\delta ]}^{-1}({y}_{0} - {b}_{p}) \in H + K$for every $p \in \mathbb{N}.$Since (b _p ) is bounded we obtain the contradiction 0 ∈  cl}(H + K). Thus we have that the sequence $\left (cl{D}_{n}\right )$is a decreasing sequence of nonempty closed subsets of the complete metric space (X, d), whose diameters tend to 0. By Cantor’s theorem, ${\bigcap \nolimits }_{n\in \mathbb{N}} cl{D}_{n} =\{ \overline{x}\}$for some $\overline{x} \in X$. Of course, ${x}_{n} \rightarrow \overline{x}$. Since $(({x}_{n},{y}_{n})) \subseteq \mathcal{A}$is a ≼  _H -decreasing sequence, from (H1) we get an $\overline{y} \in Y$such that $(\overline{x},\overline{y}) {\preccurlyeq }_{H}({x}_{n},{y}_{n})$for every $n \in \mathbb{N}$; $(\overline{x},\overline{y})$is the desired element. Indeed, $(\overline{x},\overline{y}) {\preccurlyeq }_{H}({x}_{0},{y}_{0})$. Let $({x}^{{\prime}},{y}^{{\prime}}) \in \mathcal{A}(\overline{x},\overline{y})$. It follows that $({x}^{{\prime}},{y}^{{\prime}}) \in \mathcal{A}({x}_{n},{y}_{n})$, and so x ^′ ∈ D _n  ⊆  cl}D _n for every n. Thus ${x}^{{\prime}} = \overline{x}$.

If Yis a separated locally convex space, the preceding result follows immediately from Corollary 11.8.

Of course, the set $\mathcal{A}\subseteq X \times Y$can be viewed as the graph of a multifunction $\Gamma : X \rightrightarrows Y$; then ${\Pr }_{X}(\mathcal{A}) = dom\Gamma$and ${\Pr }_{Y }(\mathcal{A}) = Im\Gamma$. In [6] one assumes that Γis level-closed, that is,

$$\begin{array}{rcl} L(b)& :=& \{x \in X\mid \exists y \in \Gamma (x) : y {\leq }_{K}b\} =\{ x \in X\mid b \in \Gamma (x) + K\} \\ & =& \{x \in X\mid \Gamma (x) \cap (b - K)\neq \varnothing \} \\ \end{array}$$

is closed for every b ∈ Y. 

For the nonempty set E ⊆ Ylet us set

$$BMMinE :=\{ \overline{y} \in E\mid E \cap (\overline{y} - K) =\{ \overline{y}\}\}$$

(see [7, (1.2)]); note that this set is different of the usual set

$$MinE :=\{ \overline{y} \in E\mid E \cap (\overline{y} - K) \subseteq \overline{y} + K\},$$

but they coincide if Kis pointed. As in [7, Definition 3.2], we say that $\Gamma : X \rightrightarrows Y$satisfies thelimiting monotonicity conditionat $\overline{x} \in dom\Gamma$if for every sequence ${\left (\left ({x}_{n},{y}_{n}\right )\right )}_{n\geq 1} \subseteq gph\Gamma$with (x _n ) converging to $\overline{x}$and (y _n ) being ≤  _K -decreasing, there exists $\overline{y} \in BMMin\Gamma (\overline{x})$such that $\overline{y} \leq {y}_{n}$for every n ≥ 1. As observed in [7], if Γsatisfies thelimiting monotonicity condition at $\overline{x} \in dom\Gamma$then $\Gamma (\overline{x}) \subseteq BMMin\Gamma (\overline{x}) + K$, that is, $\Gamma (\overline{x})$satisfies the domination property.

In [7, Proposition 3.3], in the case Ya Banach space, there are mentioned sufficient conditions in order that Γsatisfy thelimiting monotonicity condition at $\overline{x} \in dom\Gamma.$

When Xand Yare Banach spaces and His a singleton the next result is practically [7, Theorem 3.5].

Corollary 11.9 (Not Authentic Minimal-Point Theorem with Respect to ${\preccurlyeq }_{<Emphasis Type="Bold">\text{ {\it H}}</Emphasis>}$). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space. Let K ⊆ Y be a proper closed convex cone and H ⊆ K be a nonempty cs-complete bounded set with 0∉cl(H + K). Suppose that:

• (H3) $\Gamma : X \rightrightarrows Y$ is level-closed, satisfies the limiting monotonicity condition on domΓ

• (B3) ImΓ is quasi-bounded

Then for every (x ₀ ,y ₀ ) ∈ gphΓ there exist $\overline{x} \in dom\Gamma$ and $\overline{y} \in BMMin\Gamma (\overline{x})$ such that:

• (a) $(\overline{x},\overline{y}) {\preccurlyeq }_{H}({x}_{0},{y}_{0})$

• (b) (x,y) ∈ gphΓ $(x,y) {\preccurlyeq }_{H}(\overline{x},\overline{y})$ imply $x = \overline{x}.$

Proof.

In order to apply Theorem 11.7for $\mathcal{A} := gph\Gamma$we have only to show that $\mathcal{A}$verifies condition (H2). For this consider the sequence ${\left (({x}_{n},{y}_{n})\right )}_{n\geq 1} \subseteq \mathcal{A}$such that (y _n ) is ≤  _K -decreasing and ${x}_{n} \rightarrow \overline{x}$. Clearly, x _n  ∈ L(y ₁) for every n; since Γis level-closed, we have that $\overline{x} \in L({y}_{1}) \subseteq dom\Gamma$. Since Γsatisfies thelimiting monotonicity condition at $\overline{x}$, we find $\overline{y} \in BMMin\Gamma (\overline{x}) \subseteq \Gamma (\overline{x})$such that $\overline{y} \leq {y}_{n}$for every n. Hence (H2) holds. By Theorem 11.7there exists $(x,y) \in \mathcal{A}$such that (x, y) ≼  _H (x ₀, y ₀) and (x ^′, y ^′) ∈  gph}Γ, (x ^′, y ^′) ≼  _H (x, y) imply x ^′ = x. Set $\overline{x} := x$and take $\overline{y} \in BMMin\Gamma (\overline{x})$such that $\overline{y} {\leq }_{K}y$. By (11.29) we have that $(\overline{x},\overline{y}) {\preccurlyeq }_{H}({x}_{0},{y}_{0})$. Let now $({x}^{{\prime}},{y}^{{\prime}}) \in gph\Gamma = \mathcal{A}$with $({x}^{{\prime}},{y}^{{\prime}}) {\preccurlyeq }_{H}(\overline{x},\overline{y})$. Since $(\overline{x},\overline{y}) = (x,\overline{y}) {\preccurlyeq }_{H}(x,y)$, we have that (x ^′, y ^′) ≼  _H (x, y), and so ${x}^{{\prime}} = x = \overline{x}$. The proof is complete.

In the case when His a singleton the next result is practically [6, Theorem 1] under the supplementary hypothesis that Min}Γ(x) is compact for every x ∈ X; it seems that this condition has to be added in order that [6, Theorem 1] be true.

Corollary 11.10 (Not Authentic Minimal-Point Theorem with Respect to ${\preccurlyeq }_{{\it { <Emphasis Type="Bold">H</Emphasis>}}}$). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space. Let K ⊆ Y be a proper closed convex cone and H ⊆ K be a nonempty cs-complete bounded set with 0∉cl(H + K). Suppose that:

• (H4) $\Gamma : X \rightrightarrows Y$ is level-closed, MinΓ(x) is compact and Γ(x) ⊆ K + MinΓ(x) for every x ∈ domΓ

• (B3) ImΓ is quasi-bounded

Then for every (x ₀ ,y ₀ ) ∈ gphΓ there exist $\overline{x} \in dom\Gamma$ and $\overline{y} \in Min\Gamma (\overline{x})$ such that:

• (a) $(\overline{x},\overline{y}) {\preccurlyeq }_{H}({x}_{0},{y}_{0})$

• (b) (x,y) ∈ gphΓ,$(x,y) {\preccurlyeq }_{H}(\overline{x},\overline{y})$ imply $x = \overline{x}$

Proof.

In order to apply Theorem 11.7for $\mathcal{A} := gph\Gamma$we have only to show that $\mathcal{A}$verifies condition (H2). For this consider the sequence ${\left (({x}_{n},{y}_{n})\right )}_{n\geq 1} \subseteq \mathcal{A}$such that (y _n ) is ≤  _K -decreasing and ${x}_{n} \rightarrow \overline{x}$. As in the proof of the preceding corollary, $\overline{x} \in L({y}_{n})$for every $n \in \mathbb{N}$. Because $\Gamma (\overline{x}) \subseteq K + Min\Gamma (\overline{x})$, for every $n \in \mathbb{N}$there exists ${y}_{n}^{{\prime}}\in Min\Gamma (\overline{x})$such that y _n ^′ ≤ y _n . Because $Min\Gamma (\overline{x})$is compact, (y _n ^′) has a subnet (y _ψ(i) ^′) _i ∈ I converging to some $\overline{y} \in Min\Gamma (\overline{x})$; here $\psi : (I,\succcurlyeq ) \rightarrow \mathbb{N}$is such that for every nthere exists i _n  ∈ Iwith ψ(i) ≥ nfor i ≽ i _n . Hence y _ψ(i) ^′ ≤ y _ψ(i) ≤ y _n for i ≽ i _n , whence $\overline{y} \leq {y}_{n}$because Kis closed. Therefore, (H2) holds. By Theorem 11.7, for (x ₀, y ₀) ∈  gph}Γ, there exists $(x,y) \in \mathcal{A}$such that (x, y) ≼  _H (x ₀, y ₀) and (x ^′, y ^′) ∈  gph}Γ, (x ^′, y ^′) ≼  _H (x, y) imply x ^′ = x. Set $\overline{x} := x$and take $\overline{y} \in Min\Gamma (\overline{x})$such that $\overline{y} {\leq }_{K}y$. As in the proof of Corollary 11.9we find that $(\overline{x},\overline{y})$is the desired element. The proof is complete.

11.4.3 Minimal-Point Theorems of Isac–Tammer’s Type

Besides $F : X \times X \rightrightarrows K$considered in the preceding section we consider also ${F}^{{\prime}} : Y \times Y \rightrightarrows K$satisfying conditions (F1) and F(2), that is, 0 ∈ F ^′(y, y) for all y ∈ Yand ${F}^{{\prime}}({y}_{1},{y}_{2}) + F\prime({y}_{2},{y}_{3}) \subseteq F\prime({y}_{1},{y}_{3}) + K$for all y ₁, y ₂, y ₃ ∈ Y. Then $\Phi : Z \times Z \rightrightarrows K$with Z: = X×Y, defined by $\Phi (({x}_{1},{y}_{1}),({x}_{2},{y}_{2})) := F({x}_{1},{x}_{2}) + {F}^{{\prime}}({y}_{1},{y}_{2})$, satisfies conditions (F1) and (F2), too. As in Sect. 11.4.1we obtain that the relation ${\preccurlyeq }_{F,{F}^{{\prime}}}$defined by

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{2},{y}_{2})\Longleftrightarrow{y}_{2} \in {y}_{1} + F({x}_{1},{x}_{2}) + {F}^{{\prime}}({y}_{ 1},{y}_{2}) + K$$

is reflexive and transitive. Moreover, for x, x ₁, x ₂ ∈ Xand y ₁, y ₂ ∈ Ywe have

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{2},{y}_{2})\Longrightarrow({x}_{1},{y}_{1}) {\preccurlyeq }_{F}({x}_{2},{y}_{2})\Longrightarrow{y}_{1} {\leq }_{K}{y}_{2},$$

$$(x,{y}_{1}) {\preccurlyeq }_{F,{F}^{{\prime}}}(x,{y}_{2})\Longleftrightarrow{y}_{1} {\leq }_{K}{y}_{2}.$$

As in the preceding section, for Fsatisfying (F1)–(F3), F ^′satisfying (F1), (F2) and z ^∗from (F3) we define the partial order ${\preccurlyeq }_{F,{F}^{{\prime}},{z}^{{_\ast}}}$by

$$({x}_{1},{y}_{1}) {\preccurlyeq }_{F,{F}^{{\prime}},{z}^{{_\ast}}}({x}_{2},{y}_{2})\;\Longleftrightarrow\;\left \{\begin{array}{l} ({x}_{1},{y}_{1}) = ({x}_{2},{y}_{2})\text{ or} \\ ({x}_{1},{y}_{1}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{2},{y}_{2})\text{ and }{z}^{{_\ast}}({y}_{1}) < {z}^{{_\ast}}({y}_{2}). \end{array} \right.$$

Theorem 11.8 (Minimal-Point Theorem with Respect to ${\preccurlyeq }_{<Emphasis Type="Bold">\text{ {\it F}}</Emphasis>,{<Emphasis Type="Bold">\text{ {\it F}}</Emphasis>}^{{\prime}},{<Emphasis Type="Bold">\text{ {\it z}}</Emphasis>}^{{_\ast}}}$). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper convex cone. Let $F : X \times X \rightrightarrows K$ satisfy conditions (F1)–(F3), let ${F}^{{\prime}} : Y \times Y \rightrightarrows K$ satisfy (F1) and (F2), and let $\mathcal{A}\subseteq X \times Y$ satisfy the condition

• (H1b) For every ${\preccurlyeq }_{F,{F}^{{\prime}}}$ -decreasing sequence $(({x}_{n},{y}_{n})) \subseteq \mathcal{A}$ with ${x}_{n} \rightarrow x \in X$ there exists y ∈ Y such that $(x,y) \in \mathcal{A}$ and $(x,y) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n},{y}_{n})$ for every $n \in \mathbb{N}$.

Suppose that

• (B1) z ^∗ (from (F3)) is bounded from below on ${\Pr }_{Y }(\mathcal{A})$.

Then for every $({x}_{0},{y}_{0}) \in \mathcal{A}$ there exists an element $(\overline{x},\overline{y}) \in \mathcal{A}$ such that:

• (a) $(\overline{x},\overline{y}) {\preccurlyeq }_{F,{F}^{{\prime}},{z}^{{_\ast}}}({x}_{0},{y}_{0})$.

• (b) $(\overline{x},\overline{y})$ is a minimal element of $\mathcal{A}$ with respect to ${\preccurlyeq }_{F,{F}^{{\prime}},{z}^{{_\ast}}}$.

Proof.

It is easy to verify that ${\preccurlyeq }_{F,{F}^{{\prime}},{z}^{{_\ast}}}$is reflexive, transitive and antisymmetric. To get the conclusion one follows the lines of the proof of Theorem 11.6.

Clearly, taking F ^′ = 0 in Theorem 11.8we get Theorem 11.6. As mentioned after the proof of Theorem 11.6, this extends significantly [32, Theorem 3.10.7], keeping practically the same proof. We ask ourselves if [32, Theorem 3.10.15] could be extended to this framework, taking into account that the boundedness condition on $\mathcal{A}$in [32, Theorem 3.10.15] is much less restrictive. In [32, Theorem 3.10.15] we used a functional φ _A (defined by (11.5) and in (11.38) below) in order to prove the minimal-point theorem. Because an element k ⁰does not impose itself naturally, and we need a stronger condition on the functional φ _A even if k ⁰ ∈ K ∖ {0} ⊆  int}C, we consider an abstract K-monotone functional φ to which we impose some conditions φ _A has already.

Theorem 11.9 (Not Authentic Minimal-Point Theorem with Respect to ${\preccurlyeq }_{<Emphasis Type="Bold">\text{ {\it F}}</Emphasis>,{<Emphasis Type="Bold">\text{ {\it F}}</Emphasis>}^{{\prime}}}$). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper convex cone. Let $F : X \times X \rightrightarrows K$ satisfy conditions (F1)–(F3), let ${F}^{{\prime}} : Y \times Y \rightrightarrows K$ satisfy (F1) and (F2), and let $\mathcal{A}\subseteq X \times Y$ satisfy the condition

• (H1b) For every ${\preccurlyeq }_{F,{F}^{{\prime}}}$ -decreasing sequence $(({x}_{n},{y}_{n})) \subseteq \mathcal{A}$ with ${x}_{n} \rightarrow x \in X$ there exists y ∈ Y such that $(x,y) \in \mathcal{A}$ and $(x,y) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n},{y}_{n})$ for every $n \in \mathbb{N}$.

Assume that there exists a functional $\varphi : Y \rightarrow \overline{\mathbb{R}}$ such that

• (F4) $\quad ({x}_{1},{y}_{1}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{2},{y}_{2})\Longrightarrow\varphi ({y}_{1}) + d({x}_{1},{x}_{2}) \leq \varphi ({y}_{2})$.

Furthermore, suppose

• (B4) φ is bounded below on ${Pr}_{Y }(\mathcal{A})$.

Then for every point $({x}_{0},{y}_{0}) \in \mathcal{A}$ with $\varphi ({y}_{0}) \in \mathbb{R}$ , there exists $(\overline{x},\overline{y}) \in \mathcal{A}$ such that:

• (a) $(\overline{x},\overline{y}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{0},{y}_{0})$

• (b) $({x}^{{\prime}},{y}^{{\prime}}) \in \mathcal{A}$ ,$({x}^{{\prime}},{y}^{{\prime}}) {\preccurlyeq }_{F,{F}^{{\prime}}}(\overline{x},\overline{y})$ imply ${x}^{{\prime}} = \overline{x}$ (not authentic minimal point with respect to ${\preccurlyeq }_{F,{F}^{{\prime}}}$)

Moreover, if φ is strictly K-monotone on ${\Pr }_{Y }(\mathcal{A})$ , that is,${y}_{1},{y}_{2} {\in \Pr }_{Y }(\mathcal{A}),$ y ₂ − y ₁ ∈ K ∖{ 0}⇒φ(y₁) < φ(y₂), then

• (b’) $(\overline{x},\overline{y})$ is a minimal point of $\mathcal{A}$ with respect to ${\preccurlyeq }_{F,{F}^{{\prime}}}$ (minimal point with respect to ${\preccurlyeq }_{F,{F}^{{\prime}}}$).

Proof.

First note that from (F4) we have that φ is K-monotone. Let us construct a sequence ${(({x}_{n},{y}_{n}))}_{n\geq 0} \subseteq \mathcal{A}$as follows: Having $({x}_{n},{y}_{n}) \in \mathcal{A}$, we take $({x}_{n+1},{y}_{n+1}) \in \mathcal{A}$, $({x}_{n+1},{y}_{n+1}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n},{y}_{n})$, such that

$$\varphi ({y}_{n+1}) \leq \inf \{ \varphi (y)\mid (x,y) \in \mathcal{A},\ (x,y) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n},{y}_{n})\} + 1/(n + 1).$$

(11.37)

Of course, the sequence ((x _n , y _n )) is ${\preccurlyeq }_{F,{F}^{{\prime}}}$-decreasing, and so (y _n ) $({\subseteq \Pr }_{Y }(\mathcal{A}))$is K-decreasing. It follows that the sequence $\left (\varphi ({y}_{n})\right )$is non-increasing and bounded from below, hence convergent in ℝ. Because

$$({x}_{n+p},{y}_{n+p}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n},{y}_{n}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n-1},{y}_{n-1}),$$

using (F4) and (11.37) we get

$$d({x}_{n+p},{x}_{n}) \leq \varphi ({y}_{n}) - \varphi ({y}_{n+p}) \leq 1/n\quad \forall \,n,p \in {\mathbb{N}}^{{_\ast}}.$$

It follows that (x _n ) is a Cauchy sequence in the complete metric space (X, d), and so (x _n ) is convergent to some $\overline{x} \in X$.

By (H1b) there exists $\overline{y} \in Y$such that $(\overline{x},\overline{y}) \in \mathcal{A}$and $(\overline{x},\overline{y}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n},{y}_{n})$for every n ∈ ℕ. Let us show that $(\overline{x},\overline{y})$is the desired element. Indeed, $(\overline{x},\overline{y}) {\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{0},{y}_{0})$. Suppose that $({x}^{{\prime}},{y}^{{\prime}}) \in \mathcal{A}$is such that $({x}^{{\prime}},{y}^{{\prime}}) {\preccurlyeq }_{F,{F}^{{\prime}}}(\overline{x},\overline{y})$(${\preccurlyeq }_{F,{F}^{{\prime}}}({x}_{n},{y}_{n})$for every n ∈ ℕ). Thus $\varphi ({y}^{{\prime}}) + d({x}^{{\prime}},\overline{x}) \leq \varphi (\overline{y})$by (F4), whence

$$d({x}^{{\prime}},\overline{x}) \leq \varphi (\overline{y}) - \varphi ({y}^{{\prime}}) \leq \varphi ({y}_{ n}) - \varphi ({y}^{{\prime}}) \leq 1/n\quad \forall \,n \geq 1.$$

It follows that $d({x}^{{\prime}},\overline{x}) = \varphi (\overline{y}) - \varphi ({y}^{{\prime}}) = 0$. Hence ${x}^{{\prime}} = \overline{x}$.

Assuming that φ is strictly K-monotone, because ${y}^{{\prime}}{\leq }_{K}\overline{y}$and $\varphi (\overline{y}) - \varphi ({y}^{{\prime}}) = 0$, we have necessarily ${y}^{{\prime}} = \overline{y}$. Hence $(\overline{x},\overline{y})$is a minimal point with respect to ${\preccurlyeq }_{F,{F}^{{\prime}}}$.

Note that if C ⊆ Yis a proper closed convex cone such that $C - (K \setminus \{\mathbf{0}\}) = intC$and k ⁰ ∈ K ∖ {0} (see assumption (A2)), the functional ${\varphi }_{C} : Y \rightarrow \mathbb{R}$defined by (see (11.5))

$${\varphi }_{C}(y) :=\inf \left \{t \in \mathbb{R}\mid y \in t{k}^{0} + C\right \}$$

(11.38)

is a strictly K-monotone continuous sublinear functional (see Theorem 11.2). Moreover, if the condition

• $\quad {Pr}_{Y }(\mathcal{A}) \cap (\widetilde{y} - intC) = \varnothing$for some $\widetilde{y} \in Y$

holds, then φ : = φ _C is bounded from below on ${Pr}_{Y }(\mathcal{A})$, i.e., (B4) holds. Indeed, by Theorem 11.2, we have that $\varphi (y) + \varphi (-\widetilde{y}) \geq \varphi (y -\widetilde{ y}) \geq 0$for $y {\in \Pr }_{Y }(\mathcal{A})$, whence $\varphi (y) \geq -\varphi (-\widetilde{y})$for $y {\in \Pr }_{Y }(\mathcal{A}).$

Another example for a function φ is that defined by

$$\varphi (y) := {\varphi }_{K,{k}^{0}}(y -\widehat{ y}),$$

(11.39)

where Kis a proper convex cone, k ⁰ ∈ K ∖ {0}, and $\widehat{y} \in Y$is such that

• ${y}_{0} -\widehat{ y} \in \mathbb{R}{k}^{0} - K,{\quad \Pr }_{Y }(\mathcal{A}) \cap (\widehat{y} - K) = \varnothing$.

Then φ is K-monotone, φ(y ₀) < ∞and φ(y) ≥ 0 for every $y {\in \Pr }_{Y }(\mathcal{A})$, i.e., (B4) holds.

For both of these functions in (11.38) and (11.39) we have to impose condition (F4) in order to be used in Theorem 11.9.

Remark 11.9.

Using the function $\varphi = {\varphi }_{K,{k}^{0}}(\cdot -\widehat{ y})$(defined by (11.39)) in Theorem 11.9we can derive [41, Theorem 4.2] taking F(x ₁, x ₂) : = { d(x ₁, x ₂)k ⁰} and ${F}^{{\prime}}({y}_{1},{y}_{2}) :=\{ \epsilon \left \Vert {y}_{1} - {y}_{2}\right \Vert {k}^{0}\}$when Yis a Banach space; note that, at its turn, [41, Theorem 4.2] extends [46, Theorem 8].

11.4.4 Ekeland’s Variational Principles of Ha’s Type

The previous EVP type results correspond to Pareto optimality. Ha [37] established an EVP type result which corresponds to Kuroiwa optimality. The next result is an extension of this type of result. For its proof we use [65, Theorem 3.1] or [41, Theorem 2.2].

Theorem 11.10 (Variational Principle). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper convex cone. Let $F : X \times X \rightrightarrows K$ satisfy conditions (F1)–(F3) and $\Gamma : X \rightrightarrows Y$ be such that

• (H5) $\{x \in X\mid \Gamma (u) \subseteq \Gamma (x) + F(x,u) + K\}$ is closed for every u ∈ X.

Moreover, if

• (B5) z ^∗ (from (F3)) is bounded below on Γ(X),

then for every x ₀ ∈ domΓ there exists $\overline{x} \in X$ such that:

• (a) $\Gamma ({x}_{0}) \subseteq \Gamma (\overline{x}) + F(\overline{x},{x}_{0}) + K$

• (b) $\Gamma (\overline{x}) \subseteq \Gamma (x) + F(x,\overline{x}) + K$ implies $x = \overline{x}$

Proof.

Let us consider the relation ≼ on Xdefined by x ^′ ≼ xif $\Gamma (x) \subseteq \Gamma ({x}^{{\prime}}) + F({x}^{{\prime}},x) + K$. By our hypotheses we have that S(x) : = { x ^′ ∈ X∣x ^′ ≼ x} is closed for every x ∈ X. Note that for x ∈ X ∖  dom}Γwe have that S(x) = X, while for x ∈  dom}Γwe have that S(x) ⊆  dom}Γ. The relation ≼ is reflexive and transitive. The reflexivity of ≼ is obvious. Let x ^′ ≼ xand x ^′′ ≼ x ^′. Then $\Gamma (x) \subseteq \Gamma ({x}^{{\prime}}) + F({x}^{{\prime}},x) + K$and $\Gamma ({x}^{{\prime}}) \subseteq \Gamma ({x}^{{\prime\prime}}) + F({x}^{{\prime\prime}},{x}^{{\prime}}) + K$. Using (F2) we get

$$\Gamma (x) \subseteq \Gamma ({x}^{{\prime\prime}}) + F({x}^{{\prime\prime}},{x}^{{\prime}}) + K + F({x}^{{\prime}},x) + K \subseteq \Gamma ({x}^{{\prime\prime}}) + F({x}^{{\prime\prime}},x) + K,$$

that is, x ^′′ ≼ x. Consider

$$\varphi : X \rightarrow \overline{\mathbb{R}},\quad \varphi (x) :=\inf {z}^{{_\ast}}\left (\Gamma (x)\right ),$$

with the usual convention $\inf \varnothing := +\infty$. Clearly, $\varphi (x) \geq m :=\inf {z}^{{_\ast}}(\Gamma (X)) >-\infty$. Moreover, if x ^′ ≼ x ∈  dom}Γthen ${z}^{{_\ast}}(\Gamma (x)) \subseteq {z}^{{_\ast}}(\Gamma ({x}^{{\prime}})) + {z}^{{_\ast}}(F({x}^{{\prime}},x)) + {z}^{{_\ast}}(K)$, whence φ(x) ≥ φ(x ^′) + infz ^∗(F(x ^′, x)) ≥ φ(x ^′). 

Fix x ₀ ∈  dom}Γ. The conclusion of the theorem asserts that there exists $\overline{x} \in X$such that $\overline{x} \in S({x}_{0})$and $S(\overline{x}) =\{ \overline{x}\}$. To get this conclusion we apply [41, Theorem 2.2] or [65, Theorem 3.1]. Because (X, d) is complete and S(x) is closed for every x ∈ X, we may (and we do) assume that dom}Γ = X(otherwise we replace Xby S(x ₀)). In order to apply [41, Theorem 2.2] we have to show that $d({x}_{n},{x}_{n+1}) \rightarrow 0$provided (x _n ) _n ≥ 1 ⊆ Xis ≼ -decreasing. In the contrary case there exist δ > 0 and ${({n}_{p})}_{p\geq 1} \subseteq {\mathbb{N}}^{{_\ast}}$an increasing sequence such that $d({x}_{{n}_{p}},{x}_{{n}_{p}+1}) \geq \delta$for every p ≥ 1. Then, as seen above, $\varphi ({x}_{n}) \geq \varphi ({x}_{n+1}) +\inf {z}^{{_\ast}}\left (F({x}_{n+1},{x}_{n})\right )$, and so

$$\varphi ({x}_{{n}_{1}}) \geq \varphi \left ({x}_{{n}_{p}+1}\right ) +{ \sum \nolimits }_{l={n}_{1}}^{{n}_{p} }\inf {z}^{{_\ast}}\left (F({x}_{ l+1},{x}_{l})\right ) \geq m + p \cdot \eta (\delta )$$

with η(δ) > 0 from (F3). Letting $p \rightarrow \infty$we get a contradiction. Hence $d({x}_{n},{x}_{n+1}) \rightarrow 0$. The conclusion follows.

Note that instead of assuming S(u) to be closed for every u ∈ Xit is sufficient to have that S(u) is ≼ -lower closed, that is, for every ≼ -decreasing sequence (x _n ) ⊆ S(u) with ${x}_{n} \rightarrow x$we have that x ∈ S(u). Moreover, instead of using [41, Theorem 2.2] it is possible to give a slightly longer direct proof similar to that of Theorem 11.6(and using φ instead of z ^∗in the construction of (x _n )).

Remark 11.10.

Taking Yto be a separated locally convex space, K ⊆ Ya pointed closed convex cone and F(x, x ^′) : = { d(x, x ^′)k ⁰} with k ⁰ ∈ K ∖ { 0}, we can deduce [37, Theorem 3.1]. For this assume that Γ(X) is quasi bounded, Γ(x) + Kis closed for every x ∈ Xand Γis level-closed (or K-lsc with the terminology from [37]). Since clearly z ^∗is bounded from below on Im}Γ, in order to apply the preceding theorem we need to have that S(u) is closed for every u ∈ X; this is done in [37, Lemmma 3.2]. Below we provide another proof for the closedness of S(u). 

First, if x∉L(b) then there exists δ > 0 such that $B(x,\delta ) \cap L(b + \delta {k}^{0}) = \varnothing$. Indeed, because x∉L(b) we have that b∉Γ(x) + K, and so $b + {\delta }^{{\prime}}{k}^{0}\notin \Gamma (x) + K$, that is, x∉L(b + δ^′ k ⁰), for some δ^′ > 0 (since Γ(x) + Kis closed). Because L(b + δ^′ k ⁰) is closed, there exists δ ∈ (0, δ^′] such that $B(x,\delta ) \cap L(b + {\delta }^{{\prime}}{k}^{0}) = \varnothing$, and so $B(x,\delta ) \cap L(b + \delta {k}^{0}) = \varnothing.$

Fix u ∈ Xand take x ∈ X ∖ S(u), that is, $\Gamma (u)\not\subset \Gamma (x) + d(x,u){k}^{0} + K$. Then there exists y ∈ Γ(u) with $b := y - d(x,u){k}^{0}\notin \Gamma (x) + K$. By the argument above there exists δ^′ > 0 such that $B(x,{\delta }^{{\prime}}) \cap L(b + {\delta }^{{\prime}}{k}^{0}) = \varnothing$, that is, $y - d(x,u){k}^{0} + {\delta }^{{\prime}}{k}^{0}\notin \Gamma ({x}^{{\prime}}) + K$for every x ^′ ∈ B(x, δ^′). Taking δ ∈ (0, δ^′] sufficiently small we have that d(x ^′, u) ≥ d(x, u) − δ^′for x ^′ ∈ B(x, δ), and so $y\notin \Gamma ({x}^{{\prime}}) + d({x}^{{\prime}},u){k}^{0} + K$for every x ^′ ∈ B(x, δ), that is, B(x, δ) ∩ S(u) = ∅. 

If we assume that $\Gamma ({x}_{0})\not\subset \Gamma (x) + {k}^{0} + K$for every x ∈ X, then $\overline{x}$provided by the preceding theorem satisfies $d(\overline{x},{x}_{0}) < 1$. Indeed, in the contrary case, because $\Gamma ({x}_{0}) \subseteq \Gamma (\overline{x}) + d(\overline{x},{x}_{0}){k}^{0} + K$and $d(\overline{x},{x}_{0}){k}^{0} + K \subseteq {k}^{0} + K$, we get the contradiction $\Gamma ({x}_{0}) \subseteq \Gamma (\overline{x}) + {k}^{0} + K$. Replacing k ⁰by εk ⁰and dby λ^− 1 dfor some ε, λ > 0 we obtain exactly the statement of [37, Theorem 3.1].

In the case in which Yis just a topological vector space we have the following version of the preceding theorem under conditions similar to those in Theorem 11.7.

Theorem 11.11 (Variational Principle). 

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper closed convex cone. Let H ⊆ K be a nonempty cs-complete bounded set with 0∉cl(H + K), and $\Gamma : X \rightrightarrows Y$. If

• (H6) $\{x \in X\mid \Gamma (u) \subseteq \Gamma (x) + d(x,u)H + K\}$ is closed for every u ∈ X.

• (B6) Γ(X) is quasi bounded.

then for every x ₀ ∈ domΓ there exists $\overline{x} \in X$ such that:

• (a) $\Gamma ({x}_{0}) \subseteq \Gamma (\overline{x}) + d(\overline{x},{x}_{0})H + K$

• (b) $\Gamma (\overline{x}) \subseteq \Gamma (x) + d(x,\overline{x})H + K$ implies $x = \overline{x}$

Proof.

Let B ⊆ Ybe a bounded set such that Γ(X) ⊆ B + K. 

Consider F(x, x ^′) : = d(x, x ^′)Hfor x, x ^′ ∈ X. As seen before, Fsatisfies conditions (F1) and (F2), and so the relation ≼ defined in the proof of Theorem 11.10is reflexive and transitive; moreover, by our hypotheses, S(x) : = { x ^′ ∈ X∣x ^′ ≼ x} is closed for every x ∈ X. As in the proof of Theorem 11.10we may (and do) assume that X =  dom}Γand it is sufficient to show that $d({x}_{n},{x}_{n+1}) \rightarrow 0$provided (x _n ) _n ≥ 1 ⊆ Xis ≼ -decreasing. In the contrary case there exist δ > 0 and (n _p ) _p ≥ 1 ⊆ ℕ ^∗an increasing sequence such that $d({x}_{{n}_{p}},{x}_{{n}_{p}+1}) \geq \delta$for every p ≥ 1. 

Fixing y ₁ ∈ Γ(x ₁), inductively we find the sequences (y _n ) _n ≥ 0 ⊆ Y, (h _n ) _n ≥ 0 ⊆ Hand (k _n ) _n ≥ 0 ⊆ Ksuch that ${y}_{n} = {y}_{n+1} + d({x}_{n},{x}_{n+1}){h}_{n} + {k}_{n}$for every n ≥ 1. Using the convexity of H, and the facts that H ⊆ Kand Γ(X) ⊆ B + K, for p ∈ ℕwe get h _p ^′ ∈ H, b _p  ∈ Band k _p ^′, k _p ^′′ ∈ Ksuch that

$${y}_{1} = {y}_{{n}_{p+1}}+{\sum \nolimits }_{l=1}^{{n}_{p} }d({x}_{l},{x}_{l+1}){h}_{l}+{\sum \nolimits }_{l=1}^{{n}_{p} }{k}_{l} = {b}_{p}+\delta ({h}_{{n}_{1}}+\ldots +{h}_{{n}_{p}})+{k}_{p}^{{\prime}} = {b}_{ p}+p\delta {h}_{p}^{{\prime}}+{k}_{ p}^{{\prime\prime}}.$$

It follows that ${(p\delta )}^{-1}({y}_{1} - {b}_{p}) \in H + K$for every p ≥ 1. Since (b _p ) is bounded we obtain the contradiction 0 ∈  cl}(H + K). The conclusion follows.

Again, instead of assuming that S(u) is closed for every u ∈ X, it is sufficient to assume that S(u) is ≼ -lower closed for u ∈ X. A slightly longer direct proof, similar to that of Theorem 11.7, is possible. Also Theorem 11.11covers [37, Theorem 3.1].

11.4.5 Ekeland’s Variational Principle for Bi-Multifunctions

In [9] Bianchi, Kassay and Pini obtained an EVP type result for vector functions of two variables; previously such results were obtained by Isac [45] and Li et al. [49]. The next result extends [9, Theorem 1] in two directions: dis replaced by Fsatisfying (F1)–(F3) and instead of a single-valued function $f : X \times X \rightarrow Y$we take a multi-valued one. For its proof we use again [65, Theorem 3.1] or [41, Theorem 2.2].

Theorem 11.12.

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper convex cone. Let $F : X \times X \rightrightarrows K$ satisfy conditions (F1)–(F3). Assume that $G : X \times X \rightrightarrows Y$ has the properties:

• (i) 0∈ G(x,x) for every x ∈ X

• (ii) $G({x}_{1},{x}_{2}) + G({x}_{2},{x}_{3}) \subseteq G({x}_{1},{x}_{3}) + K$ for all x ₁ ,x ₂ ,x ₃ ∈ X

If

• (H7) $\left \{{x}^{{\prime}}\in X\mid \left [G(x,{x}^{{\prime}}) + F(x,{x}^{{\prime}})\right ] \cap (-K)\neq \varnothing \right \}$ is closed for every x ∈ X

• (B7) z ^∗ (from (F3)) is bounded below on the set ImG(x,⋅) for every x ∈ X

then for every x ₀ ∈ X there exists $\overline{x} \in X$ such that:

• (a) $\left [G({x}_{0},\overline{x}) + F({x}_{0},\overline{x})\right ] \cap (-K)\neq \varnothing$

• (b) $\left [G(\overline{x},x) + F(\overline{x},x)\right ] \cap (-K)\neq \varnothing$ implies $x = \overline{x}$

Proof.

Let us consider the relation ≼ on Xdefined by

$$x \preccurlyeq {x}^{{\prime}}\;\Longleftrightarrow\;\left [G({x}^{{\prime}},x) + F({x}^{{\prime}},x)\right ] \cap (-K)\neq \varnothing.$$

Then ≼ is reflexive and transitive. The reflexivity is immediate from (i) and (F1). Assume that x ≼ x ^′and x ^′ ≼ x ^′′. Then $-k \in G({x}^{{\prime}},x) + F({x}^{{\prime}},x)$and $-{k}^{{\prime}}\in G({x}^{{\prime\prime}},{x}^{{\prime}}) + F({x}^{{\prime\prime}},{x}^{{\prime}})$with k, k ^′ ∈ K. Hence, by (ii) and (F2),

$$-k-{k}^{{\prime}}\in G({x}^{{\prime}},x)+F({x}^{{\prime}},x)+G({x}^{{\prime\prime}},{x}^{{\prime}})+F({x}^{{\prime\prime}},{x}^{{\prime}}) \subseteq G({x}^{{\prime\prime}},x)+K+F({x}^{{\prime\prime}},x)+K,$$

whence $\left [G({x}^{{\prime\prime}},x) + F({x}^{{\prime\prime}},x)\right ] \cap (-K)\neq \varnothing$, that is, x ≼ x ^′′. 

Setting S(x) : = { x ^′ ∈ X∣x ^′ ≼ x}, by (H7) we have that S(x) is closed for every x ∈ X. We have to show that for (x _n ) _n ≥ 1 ⊆ Xa ≼ -decreasing sequence one has $d({x}_{n},{x}_{n+1}) \rightarrow 0$. In the contrary case there exist an increasing sequence ${({n}_{l})}_{l\geq 1} \subseteq \mathbb{N}$and δ > 0 such that $d({x}_{{n}_{l}},{x}_{{n}_{l}+1}) \geq \delta$for every l ≥ 1. Because (x _n ) is ≼ -decreasing, we have that $-{k}_{n} \in G({x}_{n},{x}_{n+1}) + F({x}_{n},{x}_{n+1})$for some k _n  ∈ Kand every n ≥ 1. Then

$$-{k}_{1} -\ldots - {k}_{n} \in G({x}_{1},{x}_{n+1}) + F({x}_{1},{x}_{2}) + \ldots + F({x}_{n},{x}_{n+1}) + K,$$

and so

$$\inf {z}^{{_\ast}}\left (ImG({x}_{ 1},\cdot )\right ) +\inf {z}^{{_\ast}}\left (F({x}_{ 1},{x}_{2})\right ) + \ldots +\inf {z}^{{_\ast}}\left (F({x}_{ n},{x}_{n+1})\right ) \leq 0\quad \forall n \geq 1.$$

Since $\inf {z}^{{_\ast}}\left (F({x}_{n},{x}_{n+1})\right ) \geq 0$for every n ≥ 1 and $\inf {z}^{{_\ast}}\left (F({x}_{{n}_{l}},{x}_{{n}_{l}+1})\right ) \geq \eta (\delta ) >0$for every l ≥ 1, taking n: = n _p with p ≥ 1, we obtain that

$$p\eta (\delta ) \leq -\inf {z}^{{_\ast}}\left (ImG({x}_{ 1},\cdot )\right )\;\mbox{ for every}\;p \geq 1.$$

This yields the contradiction η(δ) ≤ 0. Hence $d({x}_{n},{x}_{n+1}) \rightarrow 0$. Applying [41, Theorem 2.2] we get some $\overline{x} \in S({x}_{0})$with $S(\overline{x}) =\{ \overline{x}\}$, that is, our conclusion holds.

Remark 11.11.

If we need the conclusion only for a fixed (given) point x ₀ ∈ X, we may replace condition (B7) by the fact that z ^∗(from (F3)) is bounded below on the set Im}G(x ₀,  ⋅). 

Indeed, X ₀: = S(x ₀) is closed by (H7), and so (X ₀, d) is complete. If x ∈ X ₀then $-k \in G({x}_{0},x) + F({x}_{0},x) \subseteq G({x}_{0},x) + K$for some k ∈ K, and so − k ^′ ∈ G(x ₀, x) for some k ^′ ∈ K. It follows that $-{k}^{{\prime}} + G(x,u) \subseteq G({x}_{0},x) + G(x,u) \subseteq G({x}_{0},u) + K$, whence G(x, u) ⊆ G(x ₀, u) + Kfor every u ∈ X. Hence condition (B7) is verified on X ₀, and so the conclusion of the theorem holds for x ₀. 

Remark 11.12.

For F(x, x ^′) : = { d(x, x ^′)k ⁰} with k ⁰ ∈ K ∖ {0} and Gsingle-valued, using Theorem 11.12and the preceding remark one obtains [45, Theorem 8] and [49, Theorem 3]; in [45] Kis normal and closed, while in [49] k ⁰ ∈  int}K.

Note that condition (H7) in the preceding theorem holds when Gis compact-valued, G(u,  ⋅) is level-closed, Kis closed and F(x, x ^′) : = { d(x, x ^′)k ⁰} for some k ⁰ ∈ K. Indeed, assume that $-{k}_{n} \in G(u,{x}_{n}) + d({x}_{n},u){k}^{0}$for every n ≥ 1, where k _n  ∈ K. Take ε > 0. Then there exists ${n}_{\epsilon } \geq 1$such that $d({x}_{n},u) \geq d(x,u) - \epsilon =: {\gamma }_{\epsilon }$for every $n \geq {n}_{\epsilon }$. Then for such nwe have that $G(u,{x}_{n}) \cap \left (-{\gamma }_{\epsilon }{k}^{0} - K\right )\neq \varnothing$, whence $G(u,x) \cap \left (-{\gamma }_{\epsilon }{k}^{0} - K\right )\neq \varnothing$. Hence there exists y _ε ∈ G(u, x) such that ${y}_{\epsilon } + {\gamma }_{\epsilon }{k}^{0} \in -K$. Since G(u, x) is compact, (y _ε)_ε > 0has a subnet converging to y ∈ G(x, u). Since ${\lim }_{\epsilon \rightarrow 0}{\gamma }_{\epsilon } = d(x,u)$and Kis closed, we obtain that $y + d(x,u){k}^{0} \in -K.$

If Yis a separated locally convex space then we may assume that Gis weakly compact-valued instead of being compact-valued.

When Gis single-valued and F(x, x ^′) : = { d(x, x ^′)k ⁰} with k ⁰ ∈ K, where Kis closed and z ^∗(k ⁰) = 1, the preceding theorem reduces to [9, Theorem 1].

11.4.6 EVP Type Results

The framework is the same as in the previous sections. We want to apply the preceding results to obtain vectorial EVPs. To envisage functions defined on subsets of Xwe add to Yan element ∞not belonging to the space Y, obtaining thus the space Y ^∙: = Y∪{∞}. We consider that y ≤  _K ∞for all y ∈ Y. Consider now the function $f : X \rightarrow {Y }^{\bullet }$. As usual, the domain of fis domf = { x ∈ X∣f(x)≠∞}; the epigraph of fis epif = { (x, y) ∈ X×Y∣f(x) ≤  _K y}; the graph of fis gph}f = { (x, f(x))∣x ∈  epif}. Of course, fis proper if domf≠∅. For y ^∗ ∈ K ⁺we set $({y}^{{_\ast}}\circ f)(x) := +\infty$for x ∈ X ∖  domf. 

Theorem 11.13.

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper convex cone. Let $F : X \times X \rightrightarrows K$ satisfy the conditions (F1)–(F3) and let $f : X \rightarrow {Y }^{\bullet }$ be proper. Assume that

• (H8) For every sequence (x _n ) ⊆ domf with ${x}_{n} \rightarrow x \in X$ and $f({x}_{n}) \in f({x}_{n+1}) + F({x}_{n+1},{x}_{n}) + K$ for every $n \in \mathbb{N}$ one has $f({x}_{n}) \in f(x) + F(x,{x}_{n}) + K$ for every n ∈ ℕ

• (B8) z ^∗ ∘ f (with z ^∗ from (F3)) is bounded from below

Then for every x ₀ ∈ domf there exists $\overline{x} \in domf$ such that:

• (a) $f({x}_{0}) \in f(\overline{x}) + F(\overline{x},{x}_{0}) + K$

• (b) $\forall x \in domf\ :\ f(\overline{x}) \in f(x) + F(\overline{x},x) + K \Rightarrow x = \overline{x}$

Proof.

Consider $\mathcal{A} := gphf :=\{ (x,f(x))\mid x \in domf\}$. Condition (H8) says nothing than (H1) is verified. Applying Theorem 11.6we get the conclusion.

As for Theorem 11.6, in the above theorem we may assume that z ^∗is bounded from below on the set

$${B}_{0} :=\{ f(x)\mid x \in domf,\ f({x}_{0}) \in f(x) + F(x,{x}_{0}) + K\}.$$

The preceding theorem is very close to [36, Theorem 3.8] for γ = 1, which is stated for Fand Ksatisfying conditions (i), (ii), (iii), (S1), (S2) and $f : S \rightarrow Y$(with S ⊆ Xa nonempty closed set) satisfying the conditions

• (S3) Let us denote ${A}_{x}^{\gamma F} :=\{ z \in X\mid (f(z) + \gamma F(z,x)) \cap (f(x) - K)\neq \varnothing \}$for x ∈ S. For each x ∈ Sand $({z}_{n}) \subseteq {A}_{x}^{\gamma F},{z}_{n} \rightarrow z$such that f(z _n ) ≤ f(z _m ) for n > m, it follows that z ∈ A _x ^γF.

• (S4) The set $(f(S) - f({x}_{0})) \cap (-{D}_{F})$is K-bounded.

Because Sis closed one may assume that S = Xand domf = X. Observe that (S4) implies that y ^∗(B ₀) is bounded from below for every y ^∗ ∈ K ⁺, and so z ^∗(B ₀) is bounded from below. Let us prove that (S3) implies (H8) (for γ = 1). Consider (x _n ) ⊆ X =  domfwith ${x}_{n} \rightarrow x \in X$and $f({x}_{n}) \in f({x}_{n+1}) + F({x}_{n+1},{x}_{n}) + K$for every $n \in \mathbb{N}$. Clearly, for a fixed $\overline{n} \in \mathbb{N}$we have that ${({x}_{n})}_{n\geq \overline{n}} \subseteq {A}_{{x}_{\overline{n}}}^{1F}$and f(x _n ) ≤ f(x _m ) for $n \geq m \geq \overline{n}$. By (S3) we have that $x \in {A}_{{x}_{\overline{n}}}^{1F}$, that is, $f({x}_{\overline{n}}) \in f(x) + F(x,{x}_{\overline{n}}) + K$. Hence (H8) holds.

In the case in which F(x, x ^′) = d(x, x ^′)Hfor some H ⊆ Kthe condition (H8) becomes

• (H9) For every sequence (x _n ) ⊆  domfwith ${x}_{n} \rightarrow x \in X$and $f({x}_{n}) \in f({x}_{n+1}) + d({x}_{n+1},{x}_{n})H + K$for every n ∈ ℕone has $f({x}_{n}) \in f(x) + d(x,{x}_{n})H + K$for every n ∈ ℕ. 

In the case H: = { k ⁰} condition (H9) is nothing else than condition (E1) in [41].

Using Theorem 11.13and Proposition 11.7we have the following variant of the preceding result.

Theorem 11.14.

Assume that (X,d) is a complete metric space, Y is a real topological vector space and K ⊆ Y is a proper closed convex cone. Let $f : X \rightarrow {Y }^{\bullet }$ be a proper function and H ⊆ K be a nonempty cs-complete bounded set with 0∉cl(H + K). If

• (H10) For every sequence (x _n ) ⊆ domf such that ${x}_{n} \rightarrow x \in X$ and (f(x _n )) is ≤ _K -decreasing one has f(x) ≤ _K f(x _n ) for every $n \in \mathbb{N}$ .

• (B10) f(domf) is quasi bounded.

hold, then for every x ₀ ∈ domf there exists $\overline{x} \in domf$ such that:

• (a) $\left (f({x}_{0}) - K\right ) \cap \left (f(\overline{x}) + d(\overline{x},{x}_{0})H\right )\neq \varnothing$

• (b) $\left (f(\overline{x}) - K\right ) \cap \left (f(x) + d(\overline{x},x)H\right ) = \varnothing \quad \forall x \in domf \setminus \{\overline{x}\}$

Proof.

Since condition (H10) is exactly condition (H1) for $\mathcal{A} := gphf$and F = F _H , in order to have the conclusion of the theorem it is sufficient to show that (H2) is verified for this situation; then just use Proposition 11.7and Theorem 11.13.

Let $\left (({x}_{n},{y}_{n})\right ) \subseteq gphf$be such that ${x}_{n} \rightarrow x \in X$and (y _n ) is ≤  _K -decreasing. Hence y _n  = f(x _n ) for every n. By (H10) we have that $y := f(x) {\leq }_{K}f({x}_{n}) = {y}_{n}$for every $n \in \mathbb{N}$and, of course, $(x,f(x)) \in gphf$. The proof is complete.

Remark 11.13.

Taking Hto be complete, convex and bounded, then His cs-complete. In this case we obtain the main result in [8], that is, [8, Theorem 4.1].

Note that the closed convex subsets as well as the open convex subsets of a separated locally convex space are cs-closed; moreover, all the convex subsets of finite dimensional normed spaces are cs-closed (hence cs-complete).

Remark 11.14.

Taking H: = { k ⁰} in the preceding theorem one obtains practically [32, Corollary 3.10.6]; there Kis assumed to be closed in the direction k ⁰, the present condition (H10) being condition (H4) in [32, Corollary 3.10.6].

Remark 11.15.

Similar results can be stated using Theorems 11.8and 11.9. When specializing to $F({x}_{1},{x}_{2}) = \left \{d({x}_{1},{x}_{2}){k}^{0}\right \}$and ${F}^{{\prime}}({y}_{1},{y}_{2}) = \left \{\epsilon \left \Vert {y}_{1} - {y}_{2}\right \Vert {k}^{0}\right \}$one recovers [41, Corollary 3.1] and [41, Theorem 4.2].

11.5 Applications in Vector Optimization

11.5.1 Solution Concepts

Consider the vector minimization problem (VP) given as

$$V -\min f(x),\quad \text{ s.t.}\quad x \in S,$$

where Xand Yare separated locally convex spaces, {0}≠K ⊆ Yis a closed convex cone (which induces the partial order ≤  _K on Y), $f : X \rightarrow Y$and S ⊆ X. As in the preceding sections k ⁰ ∈ K ∖ ( − K) is fixed. The solution concepts for the vector optimization problem (VP) are described in the next definition.

Definition 11.1.

 

• The element y ₀ ∈ F ⊆ Yis said to be a minimal pointof Fwith respect to Kif $F \cap ({y}_{0} - K) \subseteq {y}_{0} + K$. The set of minimal points of Fwith respect to Kis denoted by Eff(F, K). An element x ₀ ∈ Sis called an efficient solutionof (VP) if f(x ₀) ∈  Eff(f(S), K).

• The element y ₀ ∈ Fis said to be a properly minimal pointof Fw.r.t. Kif there is a closed convex set A ⊆ Ywith 0 ∈  bdAand A − (K ∖ {0}) ⊆  intAsuch that $F \cap ({y}_{0} + intA) = \varnothing$. An element x ₀ ∈ Sis called a properly efficientsolution for (VP) if f(x ₀) is a properly minimal point of f(S).

• The element y ₀ ∈ Fis said to be a weakly minimal pointof Fif φK≠∅and $F \cap ({y}_{0} - intK) = \varnothing$. The set of weakly minimal points of Fis denoted by wEff(F, K). An element x ₀ ∈ Sis a weakly efficientsolutionof (VP) if f(x ₀) ∈ wEff(f(S), K).

Note first that from the very definition of weakly minimal points of Fone has

$$wEff(F,K) = F \setminus (F + intK);$$

(11.40)

then observe that the set Aappearing in the definition of a properly minimal point verifies Assumption (A2). Moreover, note that what is called here a properly minimal point of Fw.r.t. Kis said to be an E-minimal element of Fin [66] and was introduced by Iwanow and Nehse in [47] for $K = {\mathbb{R}}_{+}^{n}$and Gerstewitz and Iwanow [28] for the general case.

Lemma 11.1.

Let x ₀ ∈ S.

• (a) If x ₀ is a properly efficient solution of (VP) and A ⊆ Y is the set provided by Definition 11.1 then x ₀ is a solution of the scalar minimization problem

  $$\min \ {\varphi }_{A,{k}^{0}}(f(x) - f({x}_{0}))\quad \text{ s.t.}\quad x \in S,$$

  (11.41)

  where k ⁰ ∈ K ∖{ 0}.

• (b) If x ₀ is a weakly efficient solution of (VP) and k ⁰ ∈ intK, then x ₀ is a solution of problem(11.41) with $A := -K$.

Proof.

In both cases we have that $f(S) \cap (f({x}_{0}) + intA) = \varnothing$. Moreover, because 0 ∈  bdA, we have that φ _A (0) = 0. Assuming that ${\varphi }_{A,{k}^{0}}(f(x) - f({x}_{0})) < {\varphi }_{A,{k}^{0}}(f({x}_{0}) - f({x}_{0})) = 0$, we get the contradiction f(x) − f(x ₀) ∈  intA.

11.5.2 Necessary Optimality Conditions in Vector Optimization

We consider vector optimization problems on Asplund spaces without convexity assumptions. Recall that a Banach space Xis said to be an Asplund space (cf. Phelps [53, Definition 1.22]) if every continuous convex function defined on a nonempty open convex subset Dof Xis Fréchet differentiable at each point of some G _δsubset of D. It is known that the Banach spaces with separable dual and the reflexive Banach spaces are Asplund spaces. So c ₀and ℓ ^p, L ^p[0, 1] for 1 < p < ∞are Asplund spaces, but ℓ ¹is not an Asplund space.

Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential using the Lipschitz continuity properties of φ _A discussed in Sect. 11.3.4. In the following we provide necessary conditions for properly efficient solutions of a vector optimization problem that are related to the strong free-disposal assumption in (A2).

In order to present our results concerning the existence of Lagrange multipliers, we work with the Mordukhovich subdifferential ∂ _M and normal cone N _M (denoted ∂and Nin [51]). One says ([51, Definition 3.25]) that a function $f : X \rightarrow Y$is strictly Lipschitz at $\overline{x}$if fis Lipschitz on a neighbourhood of $\overline{x}$and there exists a neighbourhood Uof the origin in Xsuch that the sequence ${({t}_{k}^{-1}(f({x}_{k} + {t}_{k}u) - f({x}_{k})))}_{k\in \mathbb{N}}$contains a (norm) convergent subsequence whenever u ∈ U, ${x}_{k} \rightarrow \overline{x}$and t _k ↓0. It is clear that this notion reduces to local Lipschitz continuity if Yis finite dimensional.

Remark 11.16.

The function fis strictly Lipschitz at $\overline{x}$if and only if the sequence $(\Vert {u{}_{n}\Vert }^{-1}[f({x}_{n} + {u}_{n}) - f({x}_{n})])$has a norm converging subsequence whenever (x _n ) ⊆ Xconverges to $\overline{x}$, (u _n ) ⊆ X ∖ { 0} converges to 0 and the sequence ( ∥ u _n  ∥ ^− 1 u _n ) converges in X. 

For more details regarding the class of strictly Lipschitz mappings (with values in infinite dimensional spaces) see [51, Sect. 3.1.3].

We need the following calculus rules from [51] (see [51, Theorem 3.36] and [51, Corollary 3.43]) for proving one of our main results.

Lemma 11.2.

Assume that X and Y are Asplund spaces.

• (a) If f ₁ ,f ₂ : X $\rightarrow \overline{\mathbb{R}}$ are proper functions and there exists a neighbourhood U of $\overline{x} \in dom{f}_{1} \cap dom{f}_{2}$ such that f ₁ is Lipschitz on U and f ₂ is lsc on U, then

  $${\partial }_{M}({f}_{1} + {f}_{2})(\overline{x}) \subseteq {\partial }_{M}{f}_{1}(\overline{x}) + {\partial }_{M}{f}_{2}(\overline{x}).$$

• (b) If $f : X \rightarrow Y$ is strictly Lipschitz at $\overline{x}$ and $\varphi : Y \rightarrow \overline{\mathbb{R}}$ is finite and Lipschitz on a neighbourhood of $f(\overline{x})$ , then

  $${\partial }_{M}(\varphi \circ f)(\overline{x}) \subseteq \bigcup \nolimits \left \{{\partial }_{M}({y}^{{_\ast}}\circ f)(\overline{x})\mid {y}^{{_\ast}}\in {\partial }_{ M}\varphi (f(\overline{x}))\right \}.$$

In the next result we provide necessary optimality conditions for properly efficient solutions of problem (VP).

Theorem 11.15.

Assume that X and Y are Asplund spaces,$f : X \rightarrow Y$ is strictly Lipschitz, S is a closed subset of X and x ₀ ∈ S. If x ₀ is a properly efficient solution of (VP) then there exists v ^∗ ∈ K ^# such that

$$0 \in {\partial }_{M}({v}^{{_\ast}}\circ f)({x}_{ 0}) + {N}_{M}(S,{x}_{0}).$$

(11.42)

Moreover, if f is strictly differentiable at x ₀ then

$${({f}^{{\prime}}({x}_{ 0}))}^{{_\ast}}{v}^{{_\ast}}\in -{N}_{ M}(S,{x}_{0}).$$

(11.43)

Proof.

Assume that x ₀is a properly efficient solution for (VP). By Lemma 11.1, x ₀is a solution of the problem (11.41), or equivalently x ₀is a minimum point of

$$h : X \rightarrow \overline{\mathbb{R}},\quad h(x) := {\varphi }_{A}(f(x) - f({x}_{0})) + {\iota }_{S}(x),$$

where A ⊆ Yis a closed convex set such that 0 ∈  bdAand A − (K ∖ {0}) ⊆  intA. As seen in Remark 11.2(ii), domφ _A is open because k ⁰ ∈ K ∖ {0} and φ _A (0) = 0 because 0 ∈  bdA. Note that ${k}^{0}\notin {A}_{\infty }$; otherwise we get the contradiction $0 = (0 + {k}^{0}) - {k}^{0} \in A - \mathbb{P}{k}^{0} \subseteq intA$. Since 0 ∈ A ⊆  domφ _A , by Proposition 11.5we have that φ _A is convex and Lipschitz on a neighbourhood of 0. It follows that 0 ∈ ∂ _M h(x ₀) (see [51, Proposition 1.114]). Since fis strictly Lipschitz and φ _A is Lipschitz on a neighbourhood of 0, the function x↦φ _A (f(x) − f(x ₀)) is Lipschitz on a neighbourhood of x ₀. Moreover, since Sis a closed subset of Xwe have that ι _S is a proper lower-semicontinuous function. Using both parts of Lemma 11.2we have that

$$0 \in {\partial }_{M}({v}^{{_\ast}}\circ f)({x}_{ 0}) + {N}_{M}(S,{x}_{0})$$

for some ${v}^{{_\ast}}\in {\partial }_{M}{\varphi }_{A}(0) = \partial {\varphi }_{A}(0)$, φ _A being convex and finite and Lipschitz on a neighbourhood of 0. From Corollary 11.7we have that v ^∗ ∈ K ^#and $\left \langle {k}^{0},{v}^{{_\ast}}\right \rangle = 1$, and so v ^∗≠0. If fis strictly differentiable at x ₀then ∂ _M f(x ₀) = { f ^′(x ₀)}, and so the last conclusion follows.

For weakly efficient solutions of (VP) we have the following result.

Theorem 11.16.

Assume that X and Y are Asplund spaces,$f : X \rightarrow Y$ is strictly Lipschitz, S is a closed subset of X and x ₀ ∈ S. If x ₀ is a weakly efficient solution of (VP) then there exists v ^∗ ∈ K ⁺ ∖{ 0} such that (11.42) holds. Moreover, if f is strictly differentiable at x₀then (11.43) holds.

Proof.

If φK≠∅and x ₀is a weakly efficient solution for (VP), by Lemma 11.1we have that x ₀is a minimum point of hfor $A := -K$and k ⁰ ∈  intK. This time φ _A is Lipschitz and sublinear. The rest of the proof is similar.

Remark 11.17.

If Xis an Asplund space and $g : X \rightarrow \overline{\mathbb{R}}$is finite and Lipschitz on a neighbourhood of x ₀ ∈ S ⊆ Xwith Sclosed, the following well-known relations

$${\partial }_{Cl}g({x}_{0}) ={ \overline{conv}}^{{w}^{{_\ast}} }{\partial }_{M}g({x}_{0})\;\mbox{ and}\;{N}_{Cl}(S,{x}_{0}) ={ \overline{conv}}^{{w}^{{_\ast}} }{N}_{M}(S,{x}_{0})$$

hold (see [51, Theorem 3.57]), where ∂ _Cl g(x ₀) and N _Cl (S, x ₀) represent the Clarke’s subdifferential of gat x ₀and the Clarke’s normal cone of Sat x ₀, respectively. In the hypotheses of Theorem 11.15from (11.42) we get the necessary optimality condition

$$\exists {v}^{{_\ast}}\in {K}^{+} \setminus \{ 0\}\ :\ 0 \in {\partial }_{ Cl}({v}^{{_\ast}}\circ f)({x}_{ 0}) + {N}_{Cl}(S,{x}_{0})$$

(11.44)

in terms of the Clarke’s subdifferential and normal cone. However, the optimality condition given by (11.42) is sharper than the condition given by (11.44).

Remark 11.18.

Note that Theorems 11.15and 11.16remain valid when the Mordukhovich subdifferential ∂ _M is replaced by any subdifferential ∂which verifies conditions (H1)–(H4) in [17]. In such a situation Theorem 11.16corresponds to Lagrangian necessary condition for weakly efficient solutions in [17, Theorem 3.1] (compare also [20, Theorem 3.2] for the case dimY < ∞). In Theorem 11.15we have established the result for properly efficient solutions without assuming φK≠∅.

Another application envisage fuzzy necessary optimality conditions for approximate minimizers of a Lipschitz vectorial function (compare Durea and Tammer [17]). First, we need a definition.

Definition 11.2.

If α > 0 and k ⁰ ∈  intK, a point x ₀ ∈ Sis said to be (α,k ⁰ )-efficient solutionof (VP) if $(f(S) - f({x}_{0})) \cap (-\alpha {k}^{0} - K) = \varnothing$.

Of course, every weakly efficient solution of (VP) is a (α, k ⁰)-efficient solution for every α > 0 and k ⁰ ∈  intK, but the converse is false, in general.

We introduce now the concept of abstract subdifferential (see, e.g. [43]; see also [19] for a theory of subdifferentials for vector-valued functions). Let $\mathcal{X}$be a class of Banach spaces which contains the class of finite dimensional normed vector spaces. By an abstract subdifferential ∂we mean a map which associates to every lsc function $h : X \in \mathcal{X} \rightarrow \overline{\mathbb{R}}$and to every x ∈ Xa (possible empty) subset ∂h(x) ⊆ X ^∗; ∂h(x) = ∅if $f(x)\notin \mathbb{R}$. Let $X,Y \in \mathcal{X}$and denote by $\mathcal{F}(X,Y )$a class of functions acting between Xand Yhaving the property that by composition at left with a lsc function from Yto $\overline{\mathbb{R}}$the resulting function is still lsc. In the sequel we shall work in every specific case with some of the next properties of the abstract subdifferential ∂.

• (C1) If $h : X \rightarrow \overline{\mathbb{R}}$is a proper lsc convex function then ∂h(x) coincides with the Fenchel subdifferential.

• (C2) If x ∈ Xis a local minimum point for the lsc function hand $h(x) \in \mathbb{R}$then 0 ∈ ∂h(x).

  Note that (C1) and (C2) are very natural requirements for any subdifferential.

  The counterparts of “exact calculus rules” are the far more general “fuzzy calculus rules”.

• (C3) If $X \in \mathcal{X}$, $\varphi : X \rightarrow \mathbb{R}$is a locally Lipschitz functions and x ∈  domh, then

  $$\partial (h + \varphi )(x) \subseteq {\left \Vert \cdot \right \Vert }^{{_\ast}}-\limsup \limits_{ y{\rightarrow }_{ h}x,\,z\rightarrow x}(\partial h(y) + \partial \varphi (z)),$$

• (C4) If $\varphi : Y \rightarrow \mathbb{R}$is locally Lipschitz and $\psi \in \mathcal{F}(X,Y )$, then for every x,

  $$\partial (\varphi \circ \psi )(x) \subseteq {\left \Vert \cdot \right \Vert }^{{_\ast}}-\limsup \limits_{ u{\rightarrow }_{ \psi }x,\,v \rightarrow \psi (x)}{\bigcup \nolimits }_{{u}^{{_\ast}}\in \partial \varphi (v)}\partial ({u}^{{_\ast}}\circ \psi )(u).$$

where the following notations are used:

• 1. $u{\rightarrow }_{h}x$means that $u \rightarrow x$and $h(u) \rightarrow h(x);$note that if his continuous, then $u{\rightarrow }_{h}x$is equivalent with $u \rightarrow x$.

• 2. ${x}^{{_\ast}}\in {\left \Vert \cdot \right \Vert }^{{_\ast}}-{limsup}_{u\rightarrow x}\partial h(u)$means that for every ε > 0 there exist ${x}_{\epsilon }$and ${x}_{\epsilon }^{{_\ast}}$such that ${x}_{\epsilon }^{{_\ast}}\in \partial h({x}_{\epsilon })$and $\left \Vert {x}_{\epsilon } - x\right \Vert < \epsilon$, $\left \Vert {x}_{\epsilon }^{{_\ast}}- {x}^{{_\ast}}\right \Vert < \epsilon$; the notation ${x}^{{_\ast}}\in {\left \Vert \cdot \right \Vert }^{{_\ast}}-{limsup}_{u{\rightarrow }_{h}x}\partial h(u)$has a similar interpretation and is equivalent with ${x}^{{_\ast}}\in {\left \Vert \cdot \right \Vert }^{{_\ast}}-{limsup}_{u\rightarrow x}\partial h(u)$provided that his continuous.

The property (C3) is called fuzzy sum rule and a space Xon which such a property holds is called trustworthiness space for the subdifferential ∂. For example, for the Fréchet subdifferential the trustworthiness spaces are the Asplund spaces (see [23]). This rule is also satisfied (see [48, pp. 41], [18,  44] and the references therein) by:

• The proximal subdifferential when $\mathcal{X}$is the class of Hilbert spaces.

• The Fréchet subdifferential of viscosity when $\mathcal{X}$is the class of Banach spaces which admit a C ¹Lipschitz bump function.

• The β-subdifferential of viscosity when $\mathcal{X}$is the class of Banach spaces which admit a β-differentiable bump function.

The next result goes back to Durea and Tammer [17].

Theorem 11.17.

Let $X,Y \in \mathcal{X}$ , let $f \in \mathcal{F}(X,Y )$ be locally Lipschitz and let S be a closed subset of X. Let x ₀ ∈ S be a weakly efficient solution of (VP). Then for every k ⁰ ∈ intK and ε > 0 there exist u ∈ B(x ₀ ,ε), z ∈ B(x ₀ ,ε∕2) ∩ S and u ^∗ ∈ K ⁺ with u ^∗ (k ⁰ ) = 1 such that

$$0 \in \partial ({u}^{{_\ast}}\circ f)(u) + {N}_{ \partial }(S,z) + B(0,\epsilon ),$$

provided that ∂ satisfies conditions (C1), (C2), (C3), (C4). Moreover, for some x ∈ B(x ₀ ,ε∕2) and $v \in B(f(x) - f({x}_{0}),\epsilon /2)$ we have that u ^∗ (v) = φ(v).

Proof.

Let us consider ε > 0 and the functional φ given by (11.5) corresponding to a fixed k ⁰ ∈  intK. We have that

$$f({x}_{0}) \in wEff(f(S),K)$$

which means that

$$0 \in wEff(f(S) - f({x}_{0}),K).$$

Thus, φ(0) = 0 and φ(f(S) − f(x ₀)) ≥ 0, whence x ₀is a minimum point for (φ ∘ g) + ι _S , where gis defined by $g(x) = f(x) - f({x}_{0})$. From (C2) we get

$$0 \in \partial (\varphi \circ g + {\iota }_{S})({x}_{0})$$

and from (C3) (φ is Lipschitz, gis locally Lipschitz and ι _S is lsc because Sis closed), there exist x ∈ B(x ₀, ε ∕ 2), z ∈ B(x ₀, ε ∕ 2) ∩ S, p ^∗ ∈ ∂(φ ∘ g)(x), and q ^∗ ∈ N _∂ (S, z) such that

$$\left \Vert {p}^{{_\ast}} + {q}^{{_\ast}}\right \Vert < \epsilon /2.$$

Since ${p}^{{_\ast}}\in \partial (\varphi \circ g)(x)$, by (C4) there exist ${u}_{1} \in B(x,\epsilon /3) \subseteq B({x}_{0},5\epsilon /6)$, v ∈ B(g(x), ε ∕ 2), u ^∗ ∈ ∂φ(v) and v ₁ ^∗ ∈ ∂(u ^∗ ∘ g)(u ₁) such that

$$\left \Vert {v}_{1}^{{_\ast}}- {p}^{{_\ast}}\right \Vert < \epsilon /2.$$

It follows that

$$\left \Vert {v}_{1}^{{_\ast}} + {q}^{{_\ast}}\right \Vert = \left \Vert {v}_{ 1}^{{_\ast}}- {p}^{{_\ast}} + {p}^{{_\ast}} + {q}^{{_\ast}}\right \Vert < 5\epsilon /6.$$

This means that

$$0 \in \partial ({u}^{{_\ast}}\circ g)({u}_{ 1}) + {N}_{\partial }(S,z) + B(0,5\epsilon /6).$$

But

$$\partial ({u}^{{_\ast}}\circ g)({u}_{ 1}) = \partial ({u}^{{_\ast}}\circ (f(\cdot ) - f({x}_{ 0})))({u}_{1}) = \partial ({u}^{{_\ast}}\circ f)({u}_{ 1})$$

because the function u↦ − u ^∗(f(x ₀)) is constant (in particular convex). Applying (C3) we find $u \in B({u}_{1},\epsilon /6) \subseteq B({x}_{0},\epsilon )$and v ^∗ ∈ ∂(u ^∗ ∘ f)(u) such that

$$\left \Vert {v}_{1}^{{_\ast}}- {v}^{{_\ast}}\right \Vert < \epsilon /6.$$

We deduce that

$$0 \in \partial ({u}^{{_\ast}}\circ f)(u) + {N}_{ \partial }(S,z) + B(0,\epsilon ).$$

The assertions concerning u ^∗follow from Corollary 11.7and this completes the proof.

Concerning (α, k ⁰)-efficient solutions of (VP) we have the following result (compare Durea and Tammer [17]).

Theorem 11.18.

Assume that S is a closed subset of X and f is a λ-Lipschitz function. Let x ₀ ∈ S be an (α,k ⁰ )-efficient solution of (VP). Then for every e ∈ intK and ε > 0, there exist $u \in B({x}_{0},\sqrt{\alpha } + \epsilon )$ ,$z \in B({x}_{0},\sqrt{\alpha } + \epsilon /2) \cap S$ , u ^∗ ∈ K ⁺ with u ^∗ (e) = 1 and x ^∗ ∈ X ^∗ with $\left \Vert {x}^{{_\ast}}\right \Vert \leq 1$ such that

$$0 \in \partial ({u}^{{_\ast}}\circ f)(u) + \sqrt{\alpha }{u}^{{_\ast}}({k}^{0}){x}^{{_\ast}} + {N}_{ \partial }(S,z) + B(0,\epsilon ),$$

provided that ∂ satisfies conditions (C1), (C2), (C3), (C4). Moreover, for some $x \in B({x}_{0},\sqrt{\alpha } + \epsilon /2)$ and $v \in B(f(x) - f({x}_{0}),\lambda \sqrt{\alpha } + \epsilon )$ one has u ^∗ (v) = φ(v).

Proof.

Since the function fis Lipschitz, it is continuous as well, and since Sis a closed set in the Banach space X, Sis a complete metric space with respect to the metric given by the norm. Thus, it is easy to see that we are in the conditions of the vectorial variant of Ekeland principle given in Theorem 11.13. Applying this result we get an element $\overline{x} \in S$such that $\left \Vert \overline{x} - {x}_{0}\right \Vert < \sqrt{\alpha }$and having the property that it is minimal element (whence weak minimal as well) over Sfor the function hdefined by

$$h(x) := f(x) + \sqrt{\alpha }\left \Vert x -\overline{x}\right \Vert {k}^{0}.$$

Let ε > 0. One can apply now Theorem 11.17for ε replaced δ ∈ ]0, ε ∕ 2[ with $\delta (1 + \sqrt{\alpha }\left \Vert {k}^{0}\right \Vert \delta ) < 2\epsilon$. Accordingly, we can find $\overline{u} \in B(\overline{x},\delta ) \subseteq B({x}_{0},\sqrt{\alpha } + \delta )$, $x \in B(\overline{x},\delta /2) \subseteq B({x}_{0},\sqrt{\alpha } + \delta /2)$, $v \in B(h(x) - h(\overline{x}),\delta /2)$, $z \in B(\overline{x},\delta /2) \cap S \subseteq B({x}_{0},\sqrt{\alpha } + \delta /2) \cap S$and u ^∗ ∈ ∂φ(v) such that

$$0 \in \partial ({u}^{{_\ast}}\circ h)(\overline{u}) + {N}_{ \partial }(S,z) + B(0,\delta ).$$

(11.45)

Let us take the element ${\overline{x}}^{{_\ast}}\in \partial ({u}^{{_\ast}}\circ h)(\overline{u})$involved in (11.45). Since

$$\partial ({u}^{{_\ast}}\circ h)(\overline{u}) = \partial ({u}^{{_\ast}}\circ (f(\cdot ) + \sqrt{\alpha }\left \Vert \cdot -\overline{x}\right \Vert {k}^{0}))(\overline{u}),$$

by use of (C3) and (C1), there exist $u \in B(\overline{u},\delta ) \subseteq B({x}_{0},\sqrt{\alpha } + 2\delta )$and ${u}^{{\prime}}\in B(\overline{u},\delta )$such that

$${ \overline{x}}^{{_\ast}}\in \partial ({u}^{{_\ast}}\circ f)(u) + \sqrt{\alpha }{u}^{{_\ast}}({k}^{0})\partial (\left \Vert \cdot -\overline{x}\right \Vert )({u}^{{\prime}}) + B(0,\delta ).$$

(11.46)

By the calculation rule for the subdifferential of the norm and combining relations (11.45) and (11.46) it follows that there exists x ^∗ ∈ X ^∗with $\|{x}^{{_\ast}}\|\leq 1$such that

$$0 \in \partial ({u}^{{_\ast}}\circ f)(u) + \sqrt{\alpha }{u}^{{_\ast}}({k}^{0}){x}^{{_\ast}} + {N}_{ \partial }(S,z) + B(0,2\delta ).$$

Since 2δ < ε, it remains only to prove the estimation about the ball containing v. We can write

$$\begin{array}{rcl} \left \Vert v - (f(x) - f({x}_{0}))\right \Vert & \leq & \left \Vert v - (h(x) - h(\overline{x}))\right \Vert + \left \Vert (h(x) - h(\overline{x})) - (f(x) - f({x}_{0}))\right \Vert \\ & \leq & \delta /2 + \left \Vert \sqrt{\alpha }{k}^{0}\left \Vert x -\overline{x}\right \Vert - f(\overline{x}) + f({x}_{ 0}))\right \Vert \\ & \leq & \delta /2 + \sqrt{\alpha }\left \Vert {k}^{0}\right \Vert \delta /2 + \lambda \sqrt{\alpha } \\ & <& \lambda \sqrt{\alpha } + \epsilon , \\ \end{array}$$

where for the last inequality we used the assumptions made on δ. The proof is complete.