1 Introduction

Let D be a nonempty convex set in the real n-dimensional space \({\mathbb {R}}^{n}\), and let \(G: D \rightarrow {\mathbb {R}}^{n}\) be a mapping. Then one can define the variational inequality problem (VI, for short), which is to find an element \(x^{*} \in D\) such that

$$\begin{aligned} \begin{array}{c} \displaystyle \langle G(x^{*}), y - x^{*} \rangle \ge 0 \quad \forall y \in D. \end{array} \end{aligned}$$
(VI)

VIs give a suitable common format for various applied problems and are closely related with other general problems in nonlinear analysis, such as fixed point, optimization, complementarity, and equilibrium problems; see, e.g., [4,5,6,7,8,9] and the references therein. Moreover, there exist various extensions of the usual scalar VIs, in particular, vector VIs, which are closely related with vector optimization problems; see [6,7,8] for more details.

We recall that the usual vector variational inequality problem (in short, VVI) is to find an element \(x^{*} \in D\) such that

$$\begin{aligned} G(x^{*}) (y-x^{*})\notin \mathrm{-int}C \quad \forall y \in D, \end{aligned}$$
(VVI)

where G is a single-valued mapping from D into \(L({\mathbb {R}}^{n},{\mathbb {R}}^{m})\), and C is some ordering cone of \({\mathbb {R}}^{m}\) with int\(C \ne \emptyset \) (see Sect. 2). Clearly, VVI is an extension of VI in the case \(m=1\) and C is the non-negative ray. Note that each value G(x) is an \(m\times n\) matrix. However, exact values of the cost mapping G, ordering cone C, and feasible set D may be unknown for many real problems. This situation is clearly invoked by the usual calculation errors and incompleteness of information about the problem under solution. As a result, one can only deal with problems arising from suitable approximations \(\{D_k\},\) \(\{G_k\}\) and \(\{C_k\}\) of the set D,  the mapping G and the cone C,  respectively.

Our aim is to investigate convergence properties of the approximated problems, following the lines of [12, 13], where convergence of some penalty based methods for limit variational inequality problems in finite-dimensional spaces was obtained. These approximations do not require special concordance of parameters, and their convergence will be established under suitable coercivity conditions, not necessarily related to any monotonicity assumptions.

The paper is organized as follows: first, we establish a new equivalence property, which enables us to replace each VVI with a scalar set-valued variational inequality. Then, we approximate the scalar set-valued variational inequality with a sequence of penalized problems, and we study the convergence of their solutions to solutions of the original one.

2 Preliminary results

In this section we collect some preliminary notions, and some known results about VVIs. Furthermore, in view of the approximated problems considered in the sequel, we provide some known facts about generalized mixed variational inequalities. The setting is finite-dimensional; every Euclidean space will be endowed by the usual scalar product \(\langle \cdot , \cdot \rangle \) inducing the Euclidean norm \(\Vert \cdot \Vert .\) In particular, B(a, r) will denote the open ball centered at a with radius r,  and \(\overline{B(a,r)}\) its closure.

Let us recall that a nonempty set \(C\subset {\mathbb {R}}^m\) is called a convex cone if \(\lambda C\subseteq C\) for all \(\lambda > 0,\) and \(C+C=C\). A cone C is called pointed if \(C\cap (-C)=\{ \mathbf {0} \}\), where \(\mathbf {0}\) denotes the zero vector. A set X is called solid if its interior, denoted by intX,  is nonempty. Also, a nonempty set X is called proper if it is contained properly in \({\mathbb {R}}^{m}\), i.e. \(X \ne {\mathbb {R}}^{m}\). If C is a convex and proper cone, then \(\mathbf {0} \notin \mathrm{int}C\).

Given a convex, proper and solid cone, we can introduce the order \(\succ _C\) in \({\mathbb {R}}^{m}\) as follows:

$$\begin{aligned} x\succ _C y \ \Longleftrightarrow \ x-y\in \mathrm{int}C. \end{aligned}$$

In the case where

$$\begin{aligned} C={\mathbb {R}}_{+}^{m} = \{z \in {\mathbb {R}}^{m}:\, z_{i} \ge 0 i=1, \ldots ,m\}, \end{aligned}$$

we have the weak Paretian order. Note that the cone C is not supposed to be a closed set, since there are some orderings, such as the lexicographic one, whose cones are not closed.

Given a set K in \({\mathbb {R}}^{m},\) define

$$\begin{aligned} S(K)=K\cap S(\mathbf {0},1), \qquad P(K)=\mathrm{conv} S(K), \end{aligned}$$

where \(S(\mathbf {0},1)= \{ z \in {\mathbb {R}}^m: \,\Vert z\Vert =1 \},\) and \(\mathrm {conv} A\) denotes the convex hull of the set A. Furthermore, the polar (or conjugate) cone of a set K in \({\mathbb {R}}^{m}\) is given by

$$\begin{aligned} K^{*}=\{q \in {\mathbb {R}}^{m}:\, \langle p, q\rangle \ge 0, \forall p \in K\}. \end{aligned}$$

It is clear that \(K^{*}\) is a convex and closed cone. If K is itself a cone in \({\mathbb {R}}^{m},\) then \(K^{* *}=\overline{K}.\) Moreover, if the cone K is proper, convex and solid, then \(K^{*}\) has nonzero elements due to the separation theorem.

In the next lemma a characterization of the interior points of closed and convex cones is provided.

Lemma 2.1

(see e.g. [11, Lemma 1]).

  1. (i)

    Let K be a subset of \({\mathbb {R}}^{m}\). If \(p \in \mathrm{int} K\), then

    $$\begin{aligned} \langle p, q \rangle > 0\quad \forall q \in K^{*}\backslash \{ \mathbf {0}\}. \end{aligned}$$
  2. (ii)

    Let K be a convex and closed cone in \({\mathbb {R}}^{m}\). Suppose that

    $$\begin{aligned} \langle p, q \rangle > 0\quad \forall q \in K^{*}\backslash \{ \mathbf {0}\}; \end{aligned}$$

    then, \(p \in \mathrm{int} K\ (\)i.e. \(\mathrm{int} K\ne \emptyset )\).

Let us consider the following VVI: find \(x^*\in D\) such that

$$\begin{aligned} G(x^*)(y-x^*)\notin -\,\mathrm {int}(C),\quad \forall y\in D, \end{aligned}$$

where \(D\subseteq {\mathbb {R}}^n,\) and \(G=[G_1,G_2,\dots ,G_m]^{\top }\), with \(G_j:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) for every \(j=1,2,\dots , m.\)

For every element \(q \in C^{*}\backslash \{ \mathbf {0}\}\) we define the scalarized problem (\(\text {VI}_q\), for short): find a point \(x_q \in D\) such that

figure a

or, equivalently,

$$\begin{aligned} \langle G^{\top }(x_q)q,y-x_q\rangle \ge 0 \quad \forall y \in D. \end{aligned}$$

Remark 2.2

From Lemma 2.1, it is easy to prove that any solution of VVI is a solution of problem \(\text {VI}_q\), for some \(q \in C^{*}\backslash \{ \mathbf {0}\}.\) As a matter of fact, if \(x^*\) is a solution of VVI, from the convexity of D it follows that the set \(G(x^*)(D-x^*)\) is convex; moreover, it contains 0, and does not intersect \(-\,\mathrm {int}C.\) By the separation theorem, there exists \(q\in {\mathbb {R}}^n\backslash \{0\}\) such that, for all \(y\in D\) and all \(v\in -\,\mathrm {int}C,\) one has \(\langle G(x^*)(y-x^*),q\rangle >\langle v,q\rangle .\) It follows that \(\langle v, q\rangle < 0\) for all \(v\in -\,\mathrm {int}C,\) and \(\langle v, q\rangle \le 0\) for all \(v\in -\,C.\) Hence \(q \in C^*,\) and \(x^*\) is a solution of \(\text {VI}_q\). In addition, any solution of problem \(\text {VI}_q\), with \(q \in C^{*}\backslash \{ \mathbf {0}\},\) is a solution of VVI.

In order to replace the original VVI with an equivalent formulation, we define the set-valued mapping \(F: {\mathbb {R}}^n \rightarrow \Pi ({\mathbb {R}}^{n})\) as follows:

$$\begin{aligned} F(x)=\left\{ f \in {\mathbb {R}}^{n}:\,f=G^{\top }(x)q=\sum _{j=1}^m G_j(x)q_j, \ q \in P(C^{*})\right\} . \end{aligned}$$
(2.1)

Here and below \(\Pi (A)\) denotes the family of all nonempty subsets of a set A. The set-valued map F is trivially nonempty, compact and convex valued in \({\mathbb {R}}^n\), and the set \(P(C^{*})\) is compact. Under the assumption of continuity of the maps \(G_j,\) by adapting the proof of Theorem 16.34 in [3], we can easily show that F is upper semicontinuous.

Let us now consider the following generalized variational inequality problem (GVI, for short): find an element \(x^{*} \in D\) and \(f^{*} \in F(x^{*})\) such that

$$\begin{aligned} \begin{array}{c} \displaystyle \langle f^{*}, y - x^{*} \rangle \ge 0 \quad \forall y \in D. \end{array} \end{aligned}$$
(GVI)

The problem turns out to be equivalent to the problem VVI, as the next proposition shows.

Proposition 2.3

is equivalent to , where the mapping F is defined in (2.1).

Proof

If \(x^{*}\) is a solution of , then it solves problem for some \(q \in C^{*}\backslash \{ \mathbf {0}\},\) as a consequence of Lemma 2.1. Hence we can take \(q'=(1/\Vert q\Vert )q \in S(C^{*})\). This means that \(x^{*}\) is a solution of , with \(f^*=G^{\top }(x^*)q'.\)

Conversely, let \(x^{*}\) be a solution of . Then it solves for some \(q \in P(C^{*})\). By definition, there exist elements \(q^{i} \in S(C^*)\) and numbers \(\alpha _{i}> 0\) such that

$$\begin{aligned} q=\sum \limits _{i\in I} \alpha _{i} q^{i}, \qquad \sum \limits _{i\in I} \alpha _{i}=1, \end{aligned}$$

where I is a finite set of indices. Fix any point \(y \in D\). Then, there exists an index \(l\in I\) such that

$$\begin{aligned} \langle G(x^{*})(y-x^{*}),q^{l}\rangle \ge 0, \end{aligned}$$

hence, by Lemma 2.1,

$$\begin{aligned} G(x^{*}) (y-x^{*})\notin \mathrm{-int}C. \end{aligned}$$

Therefore \(x^{*}\) solves . \(\square \)

By the proposition above, existence results for VVI can be obtained by investigating the equivalent GVI. This is a special case of the more general generalized mixed variational inequality (GMVI, for short; see e.g. [13]), which is to find an element \(x^{*} \in D\) and \(f^{*} \in F(x^{*})\) such that

$$\begin{aligned} \begin{array}{c} \displaystyle \langle f^{*}, y - x^{*} \rangle + h(y)-h (x^{*})\ge 0 \quad \forall y \in D, \end{array} \end{aligned}$$
(GMVI)

where \(h: D \rightarrow {\mathbb {R}},\) and \(F: D \rightarrow \Pi ({\mathbb {R}}^{n}).\)

In the following we will consider the problem under the following basic assumptions:

(A) :

D is a nonempty, closed and convex set, \(h: D \rightarrow {\mathbb {R}}\) is a lower semicontinuous and convex function, \(F: D \rightarrow \Pi ({\mathbb {R}}^{n})\) is upper semicontinuous, with nonempty, convex, and compact values on D.

Let us recall some existence results for .

Proposition 2.4

(see [13, Proposition 2]). If A holds and D is bounded, then has a solution.

In case the set D is unbounded, some proper coercivity conditions are required. To this purpose, let us recall that a function \(\mu :{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is said to be weakly coercive with respect to a set \({\mathcal {D}}\) if there exists \(\rho \in {\mathbb {R}}\) such that the set

$$\begin{aligned} {\mathcal {D}}(\mu ,\rho ):=\{x\in {\mathcal {D}}:\, \mu (x)\le \rho \} \end{aligned}$$

is nonempty and bounded. For every \(f\in F(x),\) set

$$\begin{aligned} \Delta (f,h,x,y)= \langle f, y - x \rangle +h(y)-h (x). \end{aligned}$$

We take the following coercivity condition:

(C) :

There exist a convex function \(\mu : {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\), which is weakly coercive with respect to the set D, and a number r such that, for any point \({\bar{x}} \in D \backslash D(\mu ,r)\) and any \({\bar{f}} \in F({\bar{x}})\) with

$$\begin{aligned} \inf _{x \in D(\mu ,r)} \Delta ({\bar{f}}, h, {\bar{x}}, x) \ge 0, \end{aligned}$$
(2.2)

there is a point \(z \in D\) such that

$$\begin{aligned} \begin{array}{c} \min \{ \Delta ({\bar{f}}, h, {\bar{x}}, z), \mu (z)-\mu ({\bar{x}})\} <0, \\ \text{ and } \\ \max \{ \Delta ({\bar{f}}, h, {\bar{x}}, z), \mu (z)-\mu ({\bar{x}})\} \le 0. \end{array} \end{aligned}$$
(2.3)

The following existence result holds.

Proposition 2.5

[13, Theorem 1] If A and C are fulfilled, then has a solution.

We can somewhat strengthen the above assertion by specializing (2.3), in order to be able to localize the solutions by considering the following coercivity condition:

(\(\mathbf{C}^\prime \)):

There exist a convex function \(\mu : {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\), which is weakly coercive with respect to the set D, and a number r such that for any point \({\bar{x}} \in D \backslash D(\mu ,r)\) and any \({\bar{f}} \in F({\bar{x}})\) satisfying (2.2) there is a point \(z \in D\) such that \(\mu (z)\le \mu ({\bar{x}}) \) and

$$\begin{aligned} \Delta ({\bar{f}}, h, {\bar{x}}, z)<0. \end{aligned}$$
(2.4)

Corollary 2.6

If A and \(\mathbf{C}^\prime \) are fulfilled, then has a solution, and all the solutions are contained in \(D(\mu ,r)\).

Proof

It is enough to note that (2.4) implies (2.3), hence existence of solutions of follows from Proposition 2.5. Due to (2.4), all these solutions belong to \(D(\mu ,r)\). \(\square \)

3 Convergence of penalized approximation problems

Converting VVIs into GVIs enables us to apply the penalty approach. Following some ideas in [12, 13], we intend to establish existence results for the ‘limit’ problem by investigating the cluster points of the solutions of penalized approximation problems that are defined in terms of more regular data.

In the sequel, we will suppose that D is a set of the form

$$\begin{aligned} D=V \cap W, \end{aligned}$$
(3.1)

where V, W are convex and closed sets in the space \({\mathbb {R}}^{n}\). In general, V represents geometric constraints, whereas W corresponds to “functional” ones. To identify the points of this latter set, a suitable penalty function will be used, that is a function \(P:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) such that

$$\begin{aligned} P(w){\left\{ \begin{array}{ll} =0&{} w\in W,\\ >0&{}w\notin W. \end{array}\right. } \end{aligned}$$

We now define approximations of the data D,  G and C. Let \(\{V_k\}\) be a sequence of closed, convex subsets of \({\mathbb {R}}^n\) such that

(A1) :

\(\mathrm {Ls}\, V_k\subseteq V,\) where Ls denotes the topological limit superior, i.e.,

$$\begin{aligned} \mathrm {Ls}\,V_k=\{x\in {\mathbb {R}}^n:\, x_{n_k}\rightarrow x\;\mathrm { with}\; x_{n_k}\in V_{n_k}\}. \end{aligned}$$

The set W will be approximated via perturbed penalty functions well behaved with respect to \(\{V_k\},\) i.e., a sequence of nonnegative functions \(P_k:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) such that

  1. (𝐀𝟐)
    1. (i)

      \(P_k\) is lower semicontinuous and convex;

    2. (ii)

      if \(x_k\in V_k,\) \(x_k\rightarrow \overline{x},\) and \(\liminf P_k(x_k)=0,\) then \(P(\overline{x})=0,\) i.e., \(\overline{x}\in W;\)

    3. (iii)

      for every \(w\in V\cap W,\) there exists \(\{v_k\},\) with \(v_k\in V_k\) and \(v_k\rightarrow w\) such that \(P_k(v_k)=0\) for k large enough.

This means that the set W is approximated implicitly with a sequence \(\{W_k\}\). In addition, condition A2-(iii) admits, for some elements of the sequence \(\{W_k\}\), to be empty.

The convex, proper and solid cone C in \({\mathbb {R}}^m\) will be approximated by a sequence of convex, proper and solid cones \(\{C_k\}\) satisfying

(A3) :

\(\mathrm {Ls}\, C^*_k\subseteq C^*.\)

Denote by \(G^k\) the mapping

$$\begin{aligned} G^k=[G^k_1,G^k_2,\dots , G^k_m]^{\top }:{\mathbb {R}}^n\rightarrow L({\mathbb {R}}^n, {\mathbb {R}}^m), \end{aligned}$$

and by \(F^k\) the set-valued map \(F^k:{\mathbb {R}}^n\rightarrow \Pi ({\mathbb {R}}^n),\) defined as follows:

$$\begin{aligned} F^k(x)=\left\{ f \in {\mathbb {R}}^{n}:\, f=(G^k)^{\top }(x)q=\sum _{j=1}^m G^k_j(x)q_j, \ q \in P(C_k^{*})\right\} . \end{aligned}$$

Let us consider the following assumptions concerning the mappings \(G_j^k{:}\)

  1. (𝐀𝟒)
    1. (i)

      \(G_j^k\) is continuous, for every j, k; 

    2. (ii)

      for every sequence \(\{x_k\},\) with \(x_k\in V_k\) and \(x_k\rightarrow \overline{x},\) the set

      $$\begin{aligned} \{G^k_j(x_k),\,j=1,2,\dots ,m,\;\mathrm {and} \;k\in \mathbb {N}\} \end{aligned}$$

      is bounded;

    3. (iii)

      if \(x_k\in V_k,\) \(x_k\rightarrow \overline{x},\) \(f_k\in F^k(x_k)\) and \(f_k\rightarrow \overline{f},\) then \(\overline{f}\in F(\overline{x}).\)

Remark 3.1

Property A4-(ii) implies that the sets \(F^k(x_k)\) are uniformly bounded, for every convergent sequence \(\{x_k\}\) with \(x_k\in V_k.\)

For each k,  let us now consider the problem GMVI\(_k\): find \(x^*_k\in V_k\) and \(f^*_k\in F^k(x^*_k)\) such that

$$\begin{aligned} \langle f^*_k,y-x^*_k\rangle +\tau _k(P_k(y)-P_k(x^*_k))\ge 0,\quad \forall y\in V_k, \end{aligned}$$
(3.2)

where \(\tau _k\) is a penalty parameter such that \(\tau _k\rightarrow +\,\infty .\)

First of all, we need the following technical result:

Lemma 3.2

Suppose A3 holds. Then each limit point of any sequence \(\{q^{k}\}\), \(q^{k} \in P(C^{*}_{k}),\) belongs to \(P(C^{*})\).

Proof

Note that by A3 each limit point of the sequence \(\{ q^{k}\}\) belongs to \(C^{*}\). Without loss of generality suppose that \(q^{k} \rightarrow \bar{q}\). Then for each k there exist elements \({q}^{k,i}\in S(C^{*}_{k})\) and numbers \(\alpha _i^k \ge 0\), \(i=1, \ldots , m+1\) such that

$$\begin{aligned} q^k=\sum \limits _{i=1}^{m+1} \alpha _i^k{q}^{k,i}, \qquad \sum \limits _{i=1}^{m+1} \alpha _i^k=1. \end{aligned}$$
(3.3)

For each fixed i we have the bounded sequence \(\{{q}^{k,i}\}\). Besides, the sequence \(\{\alpha ^k\}\) with \(\alpha ^k=(\alpha _1^k, \ldots , \alpha _{m+1}^k)^{\top }\) is also bounded. Hence, taking \(m+2\) times proper subsequences, if necessary, we can suppose that \(q^{k,i} \rightarrow \bar{q}^i\) for \(i=1, \ldots , m+1\) and \(\alpha ^k \rightarrow \bar{\alpha }\), where, again by A3, \(\bar{q}^i \in S(C^{*})\) for \(i=1, \ldots , m+1\), and

$$\begin{aligned} \sum \limits _{i}^{m+1} \bar{\alpha }_{i}=1, \quad \bar{\alpha }_{i} \ge 0, i=1, \ldots , m+1. \end{aligned}$$

By (3.3),

$$\begin{aligned} \bar{q}=\sum \limits _{i=1}^{m+1} \bar{\alpha }_{i} \bar{q}^{i}, \end{aligned}$$

hence \(\bar{q} \in P(C^{*})\). \(\square \)

In order to provide existence results for (3.2), in the sequel we will assume the following coercivity condition (see, for instance, [12]):

(C1) :

for each \(k=1, 2, \dots ,\) there exist a convex function \(\mu _{k} : {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\), which is weakly coercive with respect to the set \(V_k\), and a number \(\sigma _{k}\) such that for any point \( u \in V_k\backslash V_k(\mu _{k},\sigma _{k})\) there is a point \(v \in V_k\), \(\mu _{k} (v) \le \mu _{k} (u)\) such that \( P_{k}(v) \le P_{k}(u)\) and \( G^k(u) (v-u)\in \mathrm{-\,int}C_{k}\).

Remark 3.3

(i) Condition C1 implies condition \(\mathbf{C}^\prime \), by taking \(D=V_k,\) \(r=\sigma _k,\) \({\bar{x}}=u,\) \(z=v,\) \(h=\tau _k P_k\) and \({\bar{f}}\in F_k(u).\) Indeed, take any point \(u \in V_k\backslash V_k(\mu _{k},\sigma _{k});\) then, by C1, there is \(v \in V_k\) with \(\mu _{k} (v) \le \mu _{k} (u)\) such that \( P_k(v) \le P_k(u)\) and \( G^{k}(u) (v-u)\in \mathrm{-int}C_{k}\). Then, for any \(q^k\in P(C^{*}_{k}),\) we have \(\langle q^k, G^k(u)( v - u)\rangle <0 \) due to Lemma 2.1. It follows that

$$\begin{aligned} \langle q^k, G^k(u)( v - u) \rangle +\tau _{k} (P_k(v)-P_k(u)) < 0. \end{aligned}$$

\(\mathbf{C}^\prime \) follows by setting \({\bar{f}}=\sum _{j=1}^m G_j^k(u) q_j^k.\)

(ii) In case the sets \(V_k\) are bounded, condition C1 can be trivially satisfied by taking, for instance, \(\mu _k(x)=0\) and \(\sigma _k\ge 0,\) for every k.

The following result holds:

Theorem 3.4

Let \(\{V_k\}\) and \(\{P_k\}\) satisfy A1–4. Suppose that C1 is fulfilled. Then

  1. (i)

    the problem GMVI\(_k\) has at least a solution \(x^*_k,\) and all the solutions belong to \(V_k(\mu _k,\sigma _k);\)

  2. (ii)

    any cluster point \(x^*\) of a sequence of solutions \(\{x^*_k\}\) is a solution of the problem VVI.

Proof

Assertion (i) follows directly from Corollary 2.6 and Remark 3.3-(i). Concerning (ii), by A1, we get that \(x^*\in V.\) Let us show that \(x^*\in W.\) From (3.2), there exists \(f^*_k\in F^k(x^*_k)\) such that

$$\begin{aligned} 0\le P_k(x^*_k)\le \tau _k^{-1}\langle f^*_k,y-x^*_k\rangle +P_k(y),\quad \forall y\in V_k. \end{aligned}$$
(3.4)

Take any \(w\in V\cap W;\) from A2-(iii), there exists \(\{x_k\},\) with \(x_k\in V_k\) such that \(x_k\rightarrow w\) and \(P_k(x_k)=0\) for k large enough. Therefore, by choosing \(y=x_k\) in (3.4), we get

$$\begin{aligned} 0\le P_k(x^*_k)\le \tau _k^{-1}\langle f^*_k,x_k-x^*_k\rangle . \end{aligned}$$

Denote by \(x^*_{n_k}\) a subsequence converging to \(x^*\). From assumption A4-(ii) we get that

$$\begin{aligned} 0\le \liminf P_k(x^*_{n_k})\le \lim \tau _{n_k}^{-1}\langle f^*_{n_k},x_{n_k}-x^*_{n_k}\rangle = 0, \end{aligned}$$

i.e., \(\liminf P_k(x^*_{n_k})\rightarrow 0.\) From A2-(ii), it follows that \(P(x^*)=0,\) i.e. \(x^*\in W.\) Let us prove that \(x^*\) is indeed a solution of VVI. First of all, we have already shown that \(x^* \in D\). Let \(z\in D;\) from A2-(iii) there exists \(z_k\in V_k\) such that \(z_k\rightarrow z,\) and \(P_k(z_k)=0.\) From (3.2) and the assumptions on \(P_k\) we get

$$\begin{aligned} \langle f^*_k,z_k-x^*_k\rangle \ge \tau _kP_k(x^*_k)\ge 0. \end{aligned}$$

From A4-(ii), there exists a subsequence \(\{f^*_{m_k}\}\) such that \(f^*_{m_k}\rightarrow f^*.\) It is easy to show, from A3 and A4-(iii), that \(f^*\in F(x^*).\) Taking \(k\rightarrow \infty \) in the inequality above, the assertion easily follows. \(\square \)

4 Existence results

In the previous section we focused on conditions entailing existence of solutions of the approximating problems. Moreover, we found out that any cluster points of a sequence of solutions provides a solution of the ‘limit’ problem. In order to apply Theorem 3.4, we are interested in sufficient conditions on the data leading to the existence of cluster points for any sequence of solutions of the problems \(\{\mathrm {GMVI}_k\}.\)

Following some literature on this subject (see, for instance, [12]), the first result takes into account some additional assumptions involving the weakly coercive functions \(\mu _k\) and the scalars \(\sigma _k\) associated to condition C1.

Proposition 4.1

Let C1 be satisfied for every k,  and assume that the next conditions are fulfilled:

(C2) :

If \(v_k \in V_{k}\) and \(\Vert v_k \Vert \rightarrow +\,\infty \), then \(\liminf \nolimits _{k \rightarrow \infty } \mu _{k}(v_k)\ge \sigma ''\);

(C3) :

\(\limsup \nolimits _{k \rightarrow \infty } \sigma _{k} \le \sigma ' < \sigma ''\).

Then, any sequence of solutions \(\{x^*_k\}\) has a cluster point.

Proof

From Theorem 3.4, every solution \(x^*_k\) belongs to \(V_k(\mu _k,\sigma _k),\) therefore \(\mu _k(x^*_k)\le \sigma _k.\) Assumptions C2 and C3 gives the assertion. \(\square \)

Let us now denote by \(d_H(A,B)\) the Hausdorff distance between two nonempty sets \(A,B\subset {\mathbb {R}}^m,\) defined as follows:

$$\begin{aligned} d_H(A,B):=\max \left\{ \sup _{a\in A} d(a,B),\sup _{b\in B}d(b,A)\right\} , \end{aligned}$$

where d denotes the distance endowed by the Euclidean norm. In particular, if both sets are bounded, then \(d_H(A,B)\in {\mathbb {R}}.\)

Proposition 4.2

Suppose that the following assumptions hold:

  1. (𝐁)
    1. (i)

      there exists \(\tilde{x}\in \cap _k V_k\) such that \(P_k(\tilde{x})=0;\)

    2. (ii)

      for any \(\{z_k\},\) with \(z_k\subset V_k\) and \(\Vert z_k\Vert \rightarrow +\,\infty ,\)

      $$\begin{aligned} \min _{q\in P(C^*)}\frac{\langle G^{\top }(z_k)q,z_k-\tilde{x}\rangle }{\Vert z_k-\tilde{x}\Vert }\rightarrow +\,\infty ; \end{aligned}$$
    3. (iii)

      set \(d_k(x)=d_H(F_k(x),F(x));\) if \(z_k\in V_k\) and \(\Vert z_k\Vert \rightarrow +\,\infty ,\) then \(\limsup _k d_k(z_k)\in {\mathbb {R}}.\)

Then, any sequence of solutions \(\{x^*_k\}\) admits a cluster point.

Proof

Let us argue by contradiction, by assuming that \(\Vert x^*_k-\tilde{x}\Vert \rightarrow +\,\infty .\) Since \(x^*_k\) is a solution for GMVI\(_k,\) there exists \(f^*_k\in F_k(x^*_k)\) such that

$$\begin{aligned} \langle f^*_k,y-x^*_k\rangle +\tau _k(P_k(y)-P_k(x^*_k))\ge 0,\quad \forall y\in V_k. \end{aligned}$$

Set \(y=\tilde{x};\) then, from B-(i), we get, for every k, 

$$\begin{aligned} \langle f^*_k,\tilde{x}-x^*_k\rangle \ge \tau _kP_k(x^*_k))\ge 0. \end{aligned}$$

From the definition of \(d_k\) in B-(iii), and the closedness of the set F(x) for all \(x\in D,\) there exists \(\phi _k\in F(x^*_k)\) such that

$$\begin{aligned} \Vert \phi _k-f^*_k\Vert \le d_k(x^*_k), \end{aligned}$$
(4.1)

and

$$\begin{aligned} \langle \phi _k,\tilde{x}-x^*_k\rangle +\langle f^*_k-\phi _k,\tilde{x}-x^*_k\rangle \ge 0, \end{aligned}$$

implying that

$$\begin{aligned} \frac{\langle \phi _k,x^*_k-\tilde{x}\rangle }{\Vert x^*_k-\tilde{x}\Vert } \le \frac{\langle f^*_k-\phi _k,\tilde{x}-x^*_k\rangle }{\Vert x^*_k-\tilde{x}\Vert }. \end{aligned}$$
(4.2)

Note that

$$\begin{aligned} \phi _k=G(x^*_k)^{\top }q^k,\quad \mathrm {for\; some\;}q^k\in P(C^*); \end{aligned}$$

in particular, from B-(ii), we have that

$$\begin{aligned} \frac{\langle \phi _k,x^*_k-\tilde{x}\rangle }{\Vert x^*_k-\tilde{x}\Vert }\rightarrow +\,\infty . \end{aligned}$$

As a matter of fact, from (4.1) we have that

$$\begin{aligned} \frac{\langle f^*_k-\phi _k,\tilde{x}-x^*_k\rangle }{\Vert x^*_k-\tilde{x}\Vert }\le \Vert f^*_k-\phi _k\Vert \le d_k(x^*_k). \end{aligned}$$

Therefore, from (4.2), we get

$$\begin{aligned} +\,\infty = \limsup _k d_k(x^*_k), \end{aligned}$$

contradicting the assumption B-(iii). \(\square \)

Remark 4.3

In case \(C=C^*={\mathbb {R}}^m_+,\) condition B-(ii) reduces to the usual coercivity condition for the components \(G_j\), \(j=1,2,\dots , m{:}\)

$$\begin{aligned} \frac{\langle G_j(z_k),z_k-\tilde{x}\rangle }{\Vert z_k-\tilde{x}\Vert }\rightarrow +\,\infty ,\quad \forall j=1,2,\dots ,m. \end{aligned}$$

If \(C=C_{\mathrm {lex}},\) where \(C_{\mathrm {lex}}\) denotes the cone associated to the lexicographic order, the \(C^*\) is given by

$$\begin{aligned} C^*=\{q\in {\mathbb {R}}^m:\, q=(t,0,\dots , 0),\, t\ge 0\}, \end{aligned}$$

and \(P(C^*)=\{\mathbf {e}_1\}.\) Thus, condition B-(ii) reduces to the usual coercivity condition for the first component \(G_1\) of G only:

$$\begin{aligned} \frac{\langle G_1(z_k),z_k-\tilde{x}\rangle }{\Vert z_k-\tilde{x}\Vert }\rightarrow +\,\infty . \end{aligned}$$

Under different conditions, another existence result can be stated in the framework of C-monotone maps G. Let us first recall that \(G:{\mathbb {R}}^n\subseteq {\mathbb {R}}^n\rightarrow L({\mathbb {R}}^n,{\mathbb {R}}^m)\) is said to be C-monotone on \(D\subseteq {\mathbb {R}}^n\) if

$$\begin{aligned} (G(x')-G(x))(x'-x)\in C,\quad \forall x',x\in D. \end{aligned}$$

This is equivalent to say that the map \(x\mapsto G^{\top }(x)q\) is monotone on D, for every \(q\in P(C^*).\)

In the sequel, we will suppose that the set D has nonempty interior. Denote by \(\overline{x}\) a point in \(\mathrm {int}(D).\) From Lemma 2 in [2] (see also [1]), for any \(q\in P(C^*),\) there exist positive numbers \(r_q=r_q(\overline{x})\) and \(c_q=c_q(\overline{x})\) such that

$$\begin{aligned} \langle G^{\top }(x)q, x - \overline{x} \rangle \ge r_q \Vert G^{\top }(x)q\Vert - c_q (\Vert x-\overline{x} \Vert + r_q),\quad \forall x\in D; \end{aligned}$$
(4.3)

in particular, \(c_q= \sup _{x \in \overline{B}(\overline{x}, r_q)} \Vert G^{\top }(x)q\Vert < +\,\infty \).

Proposition 4.4

Let \(\overline{x}\in \mathrm {int}(D),\) and set

$$\begin{aligned} r'=\inf _{q\in P(C^*)} r_q,\quad r''=\sup _{q\in P(C^*)} r_q, \quad c'=\sup _{q\in P(C^*)} c_q. \end{aligned}$$

Suppose that A2-(iii) holds, and the following assumptions are satisfied:

  1. (𝐁′)
    1. (i)

      the map \(G:{\mathbb {R}}^n\rightarrow L({\mathbb {R}}^n,{\mathbb {R}}^m)\) is C-monotone on D;

    2. (ii)

      for every \(\beta \in {\mathbb {R}}\) the set

      $$\begin{aligned} L_{\beta }(G):=\left\{ x\in {\mathbb {R}}^n:\, \sup _{q\in P(C^*)}\Vert G^{\top }(x)q\Vert \le \beta \Vert x\Vert \right\} \end{aligned}$$

      is bounded;

    3. (iii)

      set \(d'_k(x)=\sup _{a\in F_k(x),b\in F(x)}d(a,b);\) if \(z_k\in V_k\) and \(\Vert z_k\Vert \rightarrow +\,\infty ,\) then \(\limsup _k d'_k(z_k)\in {\mathbb {R}};\)

    4. (iv)

      \(r'>0,\) \(r'',c'\in {\mathbb {R}}.\)

Then, any sequence of solutions \(\{x^*_k\}\) admits a cluster point.

Proof

Let \(x\in D\) and \(f \in F(x).\) Then, there exists \(q\in P(C^*)\) such that \(f=G^{\top }(x)q.\) Therefore,

$$\begin{aligned} \langle f, x - \overline{x} \rangle \ge r_q \Vert f\Vert -c_q (\Vert x-\overline{x} \Vert + r_q), \end{aligned}$$

i.e.

$$\begin{aligned} r_q\Vert f\Vert \le \langle f, x - \overline{x} \rangle + c_q(\Vert x-\overline{x} \Vert + r_q). \end{aligned}$$

From \(\mathbf{B}^\prime \)-(iv),

$$\begin{aligned} r'\Vert f\Vert \le \langle f, x - \overline{x} \rangle + c'(\Vert x-\overline{x} \Vert + r''). \end{aligned}$$
(4.4)

Let now \(x^*_k\) be a solution of \(\mathrm {GMVI}_k\). Then, for some \(f^*_k \in F_k(x^*_k),\)

$$\begin{aligned} \langle f^*_k, y-x_k^* \rangle + \tau _k (P_k(y)-P_k(x^*_k)) \ge 0 \quad \forall y \in V_k. \end{aligned}$$

From assumption A2-(iii), there exist \(v_k \in V_k\), \(v_k \rightarrow \overline{x}\) such that \(P_k(v_k)=0\) for any k. Therefore, from the previous inequality, taking into account the nonnegativity of \(P_k\), we get

$$\begin{aligned} \langle f^*_k, v_k-x_k^* \rangle \ge 0. \end{aligned}$$
(4.5)

From (4.5), for any \(f_k \in F(x^*_k)\) we have

$$\begin{aligned} \langle f_k, x^*_k - \overline{x} \rangle&= \langle f_k - f^*_k, x^*_k - \overline{x} \rangle + \langle f_k^* , x^*_k - \overline{x} \rangle \\&\le \langle f_k - f^*_k, x^*_k - \overline{x} \rangle + \langle f_k^* , x^*_k - v_k \rangle + \langle f_k^* , v_k - \overline{x} \rangle \\&\le \langle f_k - f^*_k, x^*_k - \overline{x} \rangle + \langle f_k^* , v_k - \overline{x} \rangle \\&\le d'_k(x^*_k) \Vert x^*_k-\overline{x}\Vert + \langle f_k^* , v_k - \overline{x} \rangle . \end{aligned}$$

From (4.4), setting \(x=x^*_k\), we have

$$\begin{aligned} r'\Vert f_k\Vert \le \langle f_k, x^*_k - \overline{x} \rangle + c'(\Vert x^*_k-\overline{x} \Vert + r''), \end{aligned}$$

therefore

$$\begin{aligned} r'\Vert f_k\Vert&\le d'_k(x^*_k) \Vert x^*_k-\overline{x}\Vert + \langle f_k^* -f_k , v_k - \overline{x} \rangle + \\&\quad + \langle f_k , v_k - \overline{x} \rangle + c'(\Vert x^*_k-\overline{x} \Vert + r'') \\&\le (c' + d'_k(x^*_k)) \Vert x^*_k-\overline{x}\Vert + d'_k(x^*_k) \Vert v_k-\overline{x}\Vert + \Vert f_k\Vert \Vert v_k-\overline{x}\Vert . \end{aligned}$$

Since \(v_k \rightarrow \overline{x}\), there exists an integer \(k'\) such that \((r'-\Vert v_k-\overline{x}\Vert ) > r'/2\). Rearranging the terms we have:

$$\begin{aligned} r'/2 \Vert f_k\Vert \le (r'-\Vert v_k-\overline{x}\Vert ) \Vert f_k\Vert \le (c' + d'_k(x^*_k)) \Vert x^*_k-\overline{x}\Vert + d'_k(x^*_k) \Vert v_k-\overline{x}\Vert \end{aligned}$$

for any \(k \ge k'\).

Suppose that \(\Vert x^*_k\Vert \rightarrow + \infty \). Then, from \(\mathbf{B}^\prime \)-(iii), the sequence \(\{d'_k(x^*_k)\}\) is bounded from above, and therefore there exists a positive \(\beta \) such that

$$\begin{aligned} \Vert f_k\Vert \le \beta \Vert x^*_k\Vert . \end{aligned}$$

In particular, \(\sup _{q\in P(C^*)}\Vert G^{\top }(x^*_k)q\Vert \le \beta \Vert x^*_k\Vert \), i.e. \(x^*_k \in L_{\beta }(G)\), contradicting the boundedness assumption \(\mathbf{B}^\prime \)-(ii). \(\square \)