Abstract
The purpose of this paper is to study the weak subdifferential for set-valued mappings, which was introduced by Chen and Jahn (Math. Methods Oper. Res., 48:187–200, 1998). Two existence theorems of weak subgradients for set-valued mappings are obtained. Moreover, some properties of the weak subdifferential for set-valued mappings are derived. Our results improve the corresponding ones in the literature. Some examples are given to illustrate our results.
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1 Introduction
It is well known that the subgradient plays an important role in optimization and duality theory. The concept of subgradients for a convex function was considered by Rockafellar [1] in finite-dimensional spaces. In recent years, the concept of subgradients has been generalized to vector-valued mappings and set-valued mappings in abstract spaces by many authors; see [2–8]. In [9], Chen and Craven introduced the weak subgradient for a vector-valued mapping and discussed the existence of the weak subgradient. Yang [10] generalized the concept introduced by Chen and Craven [9] to set-valued mappings. Chen and Jahn [2] defined another weak subgradient, which is stronger than the weak subgradient introduced by Yang [10]. They also proved the existence of the weak subgradient by the Eidelheit separation theorem. By the Hahn–Banach theorem, Peng et al. [11] proved the existence of the weak subgradient for set-valued mappings introduced by Yang [10]. Recently, Li and Guo [12] proved some existence theorems of two kinds of weak subgradients for set-valued mappings by virtue of a Hahn–Banach extension theorem obtained by Zǎlinescu [13]. Very recently, Hernandez and Rodriguez-Marin [14] considered the weak subgradient of set-valued mappings introduced by Chen and Jahn [2] and also presented a new notion of the strong subgradient for set-valued mappings. Moreover, they obtained some existence theorems of both subgradients. Note that as mentioned above the assumptions that the cone-convexity of the objective function and the upper semicontinuity of the objective function at a given point are required. This paper is the effort in removing these restrictions.
Motivated by the work reported in [12, 14], in this paper, we consider the weak subdifferential for set-valued mappings, which was introduced by Chen and Jahn [2]. Without any convexity and upper semicontinuity assumptions on objective functions, we prove two existence theorems of weak subgradients for set-valued mappings. Moreover, we derive some properties of the weak subdifferential for set-valued mappings. Our results improve the corresponding ones in [12, 14].
2 Preliminaries
Throughout this paper, let X and Y be two real locally convex topological vector spaces, and L(X,Y) be the set of all linear continuous operators from X into Y. Let \(X':=L(X,\mathbb{R})\) and C⊂Y be a proper (i.e. {0}≠C and C≠Y) closed, convex and pointed cone with nonempty interior \(\operatorname{int}C\). The origin of X and Y are denoted by 0 X and 0 Y , respectively. Let X ∗ and Y ∗ be the topological dual spaces of X and Y, respectively. The dual cone of C is defined by
We denote by (Y,C) the ordered topological vector space, where the ordering is induced by C. For any y 1,y 2∈Y, we define the following ordering relations:
The relations > and ≯ are defined similarly.
Let F:X⇉Y be a set-valued mapping. The domain, graph and epigraph of F are, respectively, defined by
where the symbol ∅ denotes the empty set.
Let K be a nonempty subset of X, F:K⇉Y be a set-valued mapping. In this paper, we consider the following set-valued optimization problem (in short, SVOP):
A pair (x 0,y 0) with x 0∈K and y 0∈F(x 0) is called a weak efficient solution of (SVOP) iff \((F(K)-y_{0})\cap(-\operatorname{int}C)=\emptyset \), where F(K):=⋃ x∈K F(x).
Let A⊂Y. We denote by \(\operatorname{WMin}A:=\{y\in A:(A-y)\cap- \operatorname{int}C=\emptyset\}\) the set of weak efficient elements of A.
Definition 2.1
[15]
Let K be a nonempty subset of X and \(x_{0}\in\operatorname{cl}K\). The contingent cone T(K,x 0) to K at x 0 is the set of all h∈X for which there exist a net {t α :α∈I} of positive real numbers and a net {x α :α∈I}⊂K such that
Remark 2.1
From Definition 2.1, we have that \(T(K,x_{0})\subset\operatorname{clcone}(K-x_{0})\) and T(K,x 0) is a closed cone. Moreover, If K is convex, then T(K,x 0) is a closed and convex cone.
Remark 2.2
It is not difficult to see that h∈T(K,x 0) if and only if there exist a net {t α :α∈I} of positive real numbers and a net {h α :α∈I} with h α →h such that t α h α →0 and x 0+t α h α ∈K.
Definition 2.2
[3]
Let F:X⇉Y be a set-valued mapping. Let \((x_{0},y_{0})\in\operatorname{Gr}F\). The contingent derivative DF(x 0,y 0) of F at (x 0,y 0) is a set-valued mapping from X to Y defined by
Remark 2.3
Let \((x_{0},y_{0})\in\operatorname{Gr}F\). It is easy to see that
-
(i)
y∈DF(x 0,y 0)(x) if and only if there exist a net \(\{ (x_{\alpha},y_{\alpha}):\alpha\in I\}\subset\operatorname{Gr}F\) and a net {t α :α∈I} of positive real numbers such that
$$\begin{aligned} \lim_{\alpha}(x_\alpha,y_\alpha)=(x_0,y_0) \quad\mbox{and}\quad\lim_{\alpha }t_\alpha(x_\alpha-x_0,y_\alpha-y_0)=(x,y); \end{aligned}$$ -
(ii)
the set-valued mapping DF(x 0,y 0) is positively homogeneous with closed graphs;
-
(iii)
[16] \((0,0)\in\operatorname{Gr}(DF(x_{0},y_{0}))\).
Definition 2.3
[6]
Let K be a convex subset of X. A set-valued mapping F:X⇉Y is said to be C-convex on K iff, for any x 1,x 2∈K and λ∈[0,1],
Remark 2.4
If the set-valued mapping F is C-convex on K, then F(K)+C is a convex set.
Definition 2.4
[17]
A set-valued mapping F:X⇉Y is said to be compactly approximable at \((x_{0},y_{0})\in\operatorname{Gr} F\) iff, for each v 0∈X, there exists a set-valued mapping H from X into the set of all nonempty compact subsets of Y, a neighborhood V of x 0 in X, and a function r:]0,1[×X→]0,+∞) satisfying
-
(i)
\(\lim_{(t,v)\rightarrow(0^{+},v_{0})} r(t,v)=0\);
-
(ii)
for each v∈V and t∈]0,1],
$$\begin{aligned} F(x_0+tv)\subset y_0+t\bigl(H(v_0)+r(t,v) B_Y\bigr), \end{aligned}$$
where B Y is the closed unit ball around the origin of Y.
The following lemma will be used in the sequel which plays an important role in proving our main results.
Lemma 2.1
[13]
Let X, Y be separated locally convex topological vector spaces, F:X⇉Y be a C-convex set-valued mapping, X 0⊂X be a linear subspace and T 0∈L(X 0,Y). Suppose that \(\operatorname{int}(\operatorname{epi}F)\neq\emptyset\), \(X_{0}\cap\operatorname{int}(\operatorname{dom}F)\neq\emptyset\), and T 0(x)≯y for all \((x,y)\in\operatorname{Gr}F\cap(X_{0}\times Y)\). If \(T_{0} (x)=\langle x, x^{*}_{0}\rangle y_{0}\) for every x∈X 0 with fixed \(x^{*}_{0}\in X^{*}\) and y 0∈Y, then there exists T∈L(X,Y) such that \(T\mid_{X_{0}}=T_{0}\) and T(x)≯y for all \((x,y)\in\operatorname{Gr}F\).
By Lemma 2.5 in [18], it is easy to prove the following result.
Lemma 2.2
Let C⊂Y be a closed, convex and pointed cone with \(\operatorname{int}C\neq\emptyset\), and let S be a nonempty subset of Y. Then, for y∈Y,
3 Existence of Weak Subgradients
In this section, we establish two existence theorems of weak subgradients for set-valued mappings. Denote \(W:=Y\backslash(-\operatorname {int}C)\).
Definition 3.1
[2]
Let K be a subset of X with x 0∈K. Let F:K⇉Y be a set-valued mapping. T∈L(X,Y) is called a weak subgradient of F at x 0 iff
The set of all weak subgradients of F at x 0, denoted by ∂ w F(x 0), is called the weak subdifferential of F at x 0.
Theorem 3.1
Let K be a convex subset of X with \(\operatorname {int}K\neq\emptyset\). Let F:K⇉Y be a set-valued mapping with F(x)≠∅ for any x∈K. Let \(x_{0}\in\operatorname {int}K\) and \(y_{0}\in F(x_{0})\cap\operatorname{WMin}F(K)\). If the following conditions are satisfied:
-
(i)
DF(x 0,y 0) is C-convex on K−{x 0};
-
(ii)
there exists a∈Y such that \(DF(x_{0},y_{0})(K-x_{0})\subset a-\operatorname{int}C\);
-
(iii)
F(x)−F(x 0)⊂DF(x 0,y 0)(x−x 0)+C, ∀ x∈K;
then, ∂ w F(x 0)≠∅. Moreover, there exists T∈∂ w F(x 0) such that for every x∈K,
Proof
We define the set-valued mapping G:K⇉Y by
We now prove that G is a C-convex set-valued mapping. Indeed, for any x 1, x 2∈K and λ∈[0,1], by the C-convexity of DF(x 0,y 0) on K−{x 0}, we have
which implies that G is a C-convex set-valued mapping.
Let
Since K is a nonempty convex set and G is C-convex, M is a nonempty convex set. The proof of the theorem is divided into the following three steps.
(I) We prove that \(\operatorname{int}M\neq\emptyset\).
Suppose that there exists a∈Y such that
Let \(c\in\operatorname{int}C\) and y 0=a+c. Then, \(y_{0}-a=c\in\operatorname{int}C\). It follows that there exists a neighborhood U of 0 Y such that
Let \(x_{0}\in\operatorname{int}K\). Then there exists a neighborhood V of 0 X such that x 0+V⊂K. From (1), for any x∈x 0+V and y x ∈G(x), there exists \(c_{x}\in\operatorname{int}C\) such that
This fact together with (2) yields
which implies that
On the other hand, for any x∈x 0+V,
It follows that \(\operatorname{int}M\neq\emptyset\).
(II) We prove that (x 0,0)∉M.
Indeed, if (x 0,0)∈M, then \(0\in G(x_{0})+\operatorname{int}C\), and so \(G(x_{0})\cap-\operatorname{int}C\neq\emptyset\). This implies that there exists \(c\in\operatorname{int}C\) such that −c∈DF(x 0,y 0)(0). It follows that there exist nets {λ α :α∈I} of positive real numbers and \(\{(x_{\alpha},y_{\alpha}):\alpha\in I\}\subset\operatorname{Gr}F\) satisfying
Therefore, there exists α 0∈I such that
and so
which contradicts the fact \(y_{0}\in\operatorname{WMin} F(K)\).
(III) There exists T∈L(X,Y) such that T∈∂ w F(x 0).
Since M is a nonempty convex set with \(\operatorname{int}M\neq\emptyset\) and (x 0,0)∉M, by the separation theorem of convex sets, there exists (−x ∗,y ∗)∈X ∗×Y ∗∖{(0,0)} such that
or equivalently,
We claim that y ∗≠0. In fact, if y ∗=0, then 〈x ∗,x〉≤〈x ∗,x 0〉,∀x∈K. Since \(x_{0}\in\operatorname{int}K\), there exists a symmetric neighborhood U of 0 X such that x 0+U⊂K. It follows that
This implies x ∗=0, which contradicts that (−x ∗,y ∗)≠(0,0). Therefore, y ∗≠0.
Note that 0∈DF(x 0,y 0)(0). This fact together with (5) yields \(\langle y^{*},c\rangle>0,\ \forall c\in\operatorname{int}C\). And so 〈y ∗,c〉≥0, ∀c∈C, that is y ∗∈C ∗. Then there exists some \(c_{0}\in\operatorname{int}C\) with 〈y ∗,c 0〉=1. We now define a mapping T:X→Y by
Obviously, T is linear and continuous. Next we prove that for this mapping T satisfying
We now prove that
By Lemma 2.2, we only need to prove that
Suppose by contradiction that there exist x∈K and \(y\in G(x)+\operatorname {int}C\) such that
Because of y ∗∈C ∗∖{0}, we have
which contradicts (5). Therefore, by condition (iii), T∈∂ w F(x 0). Finally, for every x∈K, we have
This completes the proof. □
Remark 3.1
In [14], Hernandez and Modriguez-Marin obtained the existence theorem of weak subgradients for set-valued mappings. The assumptions that F(x 0) is upper bounded and F is upper semicontinuous at x 0 are required in [14]. However, Theorem 3.1 does not require these assumptions. The following example is given to illustrate the case that Theorem 3.1 is applicable, but Theorem 4.1 of [14] is not applicable.
Example 3.1
Let \(X=Y=\mathbb{R}\), \(K=\mathbb{R}\), C={y:y≥0}, and let
Let (x 0,y 0)=(0,0). Then,
It is easy to see that the assumptions of Theorem 3.1 are satisfied. Obviously, 0∈∂ w F(0). However, Theorem 4.1 in [14] is not applicable because F is not upper semicontinuous at x 0.
We now give a sufficient condition, which guarantees the assumption (i) in Theorem 3.1 holds.
Proposition 3.1
Let K be a convex subset of X and F:K⇉Y be a C-convex set-valued mapping. Let \((x_{0},y_{0})\in\operatorname{Gr}F\). If F is compactly approximable at (x 0,y 0), then DF(x 0,y 0) is C-convex.
Proof
Since F is compactly approximable at (x 0,y 0), by Proposition 2.2 in [19],
Since F is C-convex, \(\operatorname{epi}F\) is a convex set. It follows that \(T(\operatorname{epi}F;(x_{0},y_{0}))\) is a convex set. And so D(F+C)(x 0,y 0) is convex. Therefore, DF(x 0,y 0) is C-convex. This completes the proof. □
Theorem 3.2
Let X and Y be separated locally convex topological vector spaces, and K be a convex subset of X with \(\operatorname {int}K\neq\emptyset\). Let F:K⇉Y be a set-valued mapping with F(x)≠∅ for any x∈K. Let \(x_{0}\in\operatorname {int}K\) and y 0∈F(x 0). If the following conditions are satisfied:
-
(i)
DF(x 0,y 0) is C-convex on K−{x 0};
-
(ii)
\(DF(x_{0},y_{0})(0)\cap-\operatorname{int}C=\emptyset\);
-
(iii)
F(x)−F(x 0)⊂DF(x 0,y 0)(x−x 0)+C, ∀x∈K;
then, ∂ w F(x 0)≠∅.
Proof
Let S=K−{x 0}. We define the set-valued mapping G:X⇉Y by
Similarly to the proof of Theorem 3.1, we can prove that G is C-convex on S.
By condition (ii), \(G(0)\cap-\operatorname{int}C=\emptyset\). It follows that
We next consider the special subspace X 0={0} and T 0(0):=0. Since \(x_{0}\in\operatorname{int}K\) and F(x)≠∅ for any x∈K, it is easy to see that
From (6), we have
By Lemma 2.1, there exists T∈L(X,Y) such that
and so
It follows that
or
which, together with condition (iii), yields
This implies T∈∂ w F(x 0). This completes the proof. □
Remark 3.2
In [12, Theorem 3.2], Li and Guo obtained a existence theorem of weak subgradients by using similar proof methods. It is important to note that our assumptions are different from the ones used in [12]. First, the condition that F is C-convex has been relaxed because we consider C-convexity of the contingent derivative of F instead of F. Second, the assumptions that F(x 0)−C is convex and \(F(x_{0})\cap(F(x_{0})-\operatorname {int}C)=\emptyset\) are required in [12], but Theorem 3.2 does not require these assumptions.
Remark 3.3
In [2, Theorem 7] and [11, Theorem 4.1], the authors derived some existence theorems of weak subgradients for set-valued mappings. The assumptions that −F(x 0) is minorized, F is C-convex and upper semicontinuous at x 0 are required in [2, 11]. However, Theorem 3.2 does not require these assumptions.
Now, we give an example to illustrate Theorem 3.2.
Example 3.2
Let \(X=Y=\mathbb{R}\), \(K=\mathbb{R}\), C={y:y≥0}, and let
Let (x 0,y 0)=(0,0). Then,
It is easy to see that the assumptions of Theorem 3.2 are satisfied. Obviously, 0∈∂ w F(0). However, Theorem 3.2 in [12], Theorem 7 in [2] and Theorem 4.1 in [11] are not applicable since F is not C-convex on K. Indeed, letting x 1=−4, x 2=2 and \(\lambda=\frac{1}{2}\), we have
4 Properties of Weak Subgradients
In this section, we obtain some properties of weak subgradients for set-valued mappings.
Theorem 4.1
Let K be a convex subset of X and x 0∈K. Let F:K⇉Y be a C-convex set-valued mapping with nonempty values and F(x 0)−C is convex. If T∈∂ w F(x 0), then there exists y ∗∈C ∗∖{0} such that
Proof
Let T∈∂ w F(x 0). Then
This implies that
We define a set-valued mapping G:K⇉Y by
Since F is C-convex and F(x 0)−C is convex, for any x 1, x 2∈K and λ∈[0,1],
It follows that G is a C-convex set-valued mapping. Note that T is a linear operator, then
is a convex set. By the separation theorem of convex sets, there exists y ∗∈Y ∗∖{0} such that
We claim that
In fact, if there exists c 0∈C such that 〈y ∗,c 0〉<0, then by letting x=x 0 and y=y 1 in (7), we have 〈y ∗,c 0〉≥0. This gives a contradiction. Thus, y ∗∈C ∗∖{0}. Letting c=0 in (7), we get the conclusion. This completes the proof. □
Theorem 4.2
Let K be a nonempty subset of X and x 0∈K. Let F:K⇉Y be a set-valued mapping with nonempty values. If ∂ w F(x 0)≠∅, then ∂ w F(x 0) is a closed set.
Proof
Suppose by contradiction that there exists a net {T α :α∈I}⊂∂ w F(x 0) such that T α →T, but T∉∂ w F(x 0). Thus, there exist \(\overline{x}\in K\), \(\overline{y}\in F(\overline{x})\) and y 0∈F(x 0) such that
Note that
It follows that there exists α 0∈I such that
which contracts the fact T α ∈∂ w F(x 0). This completes the proof. □
Remark 4.1
Note that we prove Theorem 4.2 in locally convex topological vector spaces. But a similar result has been proved by Li and Guo [12] in normed spaces.
5 Conclusions
In this paper, we have proved two existence theorems of weak subgradients for set-valued mappings. These two results improve meaningfully the corresponding results obtained by Hernandez and Rodriguez-Marin [14] and Li and Guo [12], respectively. Moreover, two properties of the weak subdifferential for set-valued mappings are derived. It would be interesting to consider the calculations of sum mapping and composed mapping for weak subdifferentials as well as applications to set-valued optimization problems. This may be the topic of some of our forthcoming papers.
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Acknowledgements
The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11001287, 11171363 and 11201509), the Natural Science Foundation Project of Chongqing (CSTC 2010BB9254 and CSTC 2009BB8240), the Education Committee Project Research Foundation of Chongqing (KJ100711), the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01) and the project of the third batch support program for excellent talents of Chongqing City High Colleges.
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Communicated by Jafar Zafarani.
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Long, X.J., Peng, J.W. & Li, X.B. Weak Subdifferentials for Set-Valued Mappings. J Optim Theory Appl 162, 1–12 (2014). https://doi.org/10.1007/s10957-013-0469-3
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DOI: https://doi.org/10.1007/s10957-013-0469-3