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The existence of contingent epiderivative for a setvalued mapping and vector variationallike inequalities
Journal of Inequalities and Applications volume 2013, Article number: 352 (2013)
Abstract
In this paper we analyze the existence of the contingent epiderivative for a setvalued mapping in a general normed space with respect to Daniell cone. As its application, we will also establish the relationships between setvalued optimization problems and variationallike inequality problems under the conditions of pseudo invexity.
MSC:49J52, 90C56.
1 Introduction
In [1], Jahn and Raüh introduced the notion of contingent epiderivative for a setvalued mapping, which modifies a notion introduced by Aubin [2] as upper contingent derivative, and also established the existence theory of this contingent epiderivative for a singlevalued function. In [3], Jahn and Khan obtained the existence of this kind of contingent epiderivative for a real setvalued function. It has been shown that this notion of contingent epiderivative is a fundamental concept for the formulation of optimality conditions in setvalued optimization, but there are few works that study its existence for a setvalued mapping in general conditions. Although in [4] RodríguezMarín and Sama derived the existence of contingent epiderivative, this can only be assured if a setvalued mapping F has the LBD (lower bounded derivative) property. In the last decades, many researchers have given several other generalized notions of epiderivatives by using weak minimizers and minimizers and derived the existence theories for them; see [5] and [4, 6, 7], respectively. Using different kinds of minimal elements, one can define different kinds of epiderivatives. In our paper, we use the ideal minimal elements of a set and the concept of contingent cone to define the contingent epiderivative and analyze its domain, existence, uniqueness and other properties. Under determined conditions, we establish dom(DF({x}_{0},{y}_{0}))=cone(A{x}_{0})=T(A,{x}_{0}) and get the existence of DF({x}_{0},{y}_{0}), where domF is the domain of F, DF({x}_{0},{y}_{0}) is the contingent epiderivative of F at ({x}_{0},{y}_{0})\in grF (the graph of F is denoted by grF) and T(A,{x}_{0}) is the contingent cone of A at {x}_{0}, respectively.
Our other purpose in this paper is to investigate the relationships between setvalued optimization problems and variationallike inequality problems. In fact, the relationships between vector variationallike inequality problems and optimization problems for a singlevalued mapping have been studied by many authors, see [8–10] and so on; and in [11], Zeng and Li also discussed the relationships between weak vector variationallike inequality problems and setvalued optimization problems. However, to the best of our knowledge, there are few papers discussing the solution relationships between setvalued optimization problems and strong vector variationallike inequality problems. Motivated by the works in [9] and [11], in this paper, we firstly introduce several kinds of generalized invexity for setvalued mappings and then prove that the solutions of the variationallike inequality problems are equivalent to the minima (ideal minima) of setvalued optimization problems.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries and recall the main notions of contingent cone. In Section 3, the concept of contingent epiderivatives is introduced and, under determined conditions, its existence theory is also established. In Section 4, we present that the solutions of the variationallike inequality problems are equivalent to the minima (ideal minima) of setvalued optimization problems.
2 Preliminaries and notations
Throughout the paper, if not stated otherwise, let X be a real normed space, let A be a nonempty subset of X, and let Y be a real normed space partially ordered by a closed, convex and pointed cone D\subset Y. The points of origin of all real normed spaces are denoted by {0}_{X} and {0}_{Y}.
Let {y}_{1},{y}_{2}\in Y, the orderings are defined in Y as follows:
{y}_{1}\nleqq {y}_{2}\iff {y}_{2}{y}_{1}\notin D;
{y}_{1}\le {y}_{2}\iff {y}_{2}{y}_{1}\in D.
Let A\subset X, F:A\to {2}^{Y} be a setvalued mapping. The graph, the epigraph and the domain of F are defined, respectively, by
and
We say that F is a Dconvex setvalued mapping on A, if A is a convex set, and for all x,y\in A, all \lambda \in [0,1],
It is well known that if F is Dconvex on A, then epiF is a convex subset in X\times Y.
Proposition 2.1 (see [12])
Let F:X\to {2}^{Y}. If epiF is a closed subset in X\times Y, then F(x)+D is a closed set for each x\in X.
Definition 2.1 Let B be a nonempty subset of Y.

(i)
{y}_{0}\in B is called a minimal point of B with respect to cone D if (B{y}_{0})\cap D=\{{0}_{{}_{Y}}\};

(ii)
{y}_{0}\in B is called an ideal minimal point of B with respect to cone D if B{y}_{0}\subseteq D.
The elements of all minimal points and ideal minimal points of B are denoted by MinB and IMinB, respectively.
Obviously, IMinB\subseteq MinB if D is a pointed cone.
The following standard notions can be found in [6].
Definition 2.2 Let D be a closed, convex and pointed cone in Y.

(i)
The cone D is called Daniell cone if any decreasing sequence in Y having a lower bound converges to its infimum.

(ii)
A subset B of Y is said to be Dlower bounded (or be minorized) if there is an element y\in Y such that B\subset y+D.
Proposition 2.2 Let D be a closed, convex and pointed cone in Y, nonempty subset B\subset Y. If MinB\ne \mathrm{\varnothing} and B is the Dlower bounded for every minimal point, i.e.,
then MinB is a singlepoint set and MinB=IMinB.
Proof We prove this result by contradiction. Suppose that at least there are {y}_{1},{y}_{2}\in MinB and {y}_{1}\ne {y}_{2}. By condition (2.1), we have
and
From (2.2), (2.3) and the assumption {y}_{1}\ne {y}_{2}, one has {y}_{1}{y}_{2}\in D\setminus \{{0}_{{}_{Y}}\} and {y}_{2}{y}_{1}\in D\setminus \{{0}_{Y}\}, which contradicts D being a pointed cone, thus {y}_{1}={y}_{2}=\overline{y}. Namely, MinB is a singleton. Again from (2.1), we have
i.e.,
this yields \overline{y}\in IMinB. The proof is complete. □
Remark 2.1 We notice if MinB\ne \mathrm{\varnothing} and (2.1) is fulfilled, then IMinB=MinB\ne \mathrm{\varnothing} is a singlepoint set. Clearly, if IMinB\ne \mathrm{\varnothing}, then B\subseteq IMinB+D.
The following theorem is due to Borwein.
Theorem 2.1 (see [13])
Assume that D is a pointed, convex and Daniell cone, and let B be a closed subset of Y. If B is Dlower bounded (or is minorized), then MinB\ne \mathrm{\varnothing}.
Definition 2.3 For A\subset X, coneA and clA denote its cone hull and closure of A, respectively.
Furthermore,
Let A\subset X, the contingent cone of A at {x}_{0} is defined by
It is well known that if A is a convex set and {x}_{0}\in A, then
We say that A\subset X satisfies the property Ϝ, if for any x\in A and \lambda \in [0,1], one has \lambda x\in A. Let {0}_{X}\in A, we say that A satisfies the property Λ near the {0}_{X}, if there exists neighborhood B({0}_{X},\epsilon ) such that for any x\in B({0}_{X},\epsilon )\cap (clA\setminus intA) and \lambda \in [0,1], one has \lambda x\in (clA\setminus intA), where B({0}_{X},\epsilon ) denotes the ball centered at {0}_{X} with radius ε.
Corollary 1 Let A\subset X be a convex set, {x}_{0}\in A, then A{x}_{0} satisfies the property Ϝ.
Proof For any x\in A and t\in [0,1], t(x{x}_{0})+{x}_{0}=tx+(1t){x}_{0}, thereby, t(x{x}_{0})+{x}_{0}\in A follows immediately from A being a convex set, which implies t(x{x}_{0})\in A{x}_{0}, the proof is complete. □
Proposition 2.3 Let A\subset X, if A satisfies the properties Ϝ and Λ near the {0}_{X}, then cone(clA)=cl(coneA).
Proof Obviously, {0}_{X}\in cone(clA) and {0}_{X}\in cl(coneA), so we consider u\ne {0}_{X} in the sequel. First, we prove the inclusion cone(clA)\subseteq cl(coneA). Let u\in cone(clA), then there exist a\in clA, t>0 such that u=ta. Furthermore, following a\in clA, there exists \{{a}_{n}\}\subset A, {a}_{n}\to a. For each n, set {u}_{n}=t{a}_{n}, then {u}_{n}\in coneA and {u}_{n}=t{a}_{n}\to ta=u, i.e., {u}_{n}\to u, this implies u\in cl(coneA).
For the contrary inclusion, we should only prove cone(clA) is a closed cone. Let \{{u}_{n}\}\subset cone(clA) and {u}_{n}\to u. Next, we will prove u\in cone(clA). In fact, we can conclude that there exist {x}_{n}\in X and {x}_{n}\to {0}_{X} such that u+{x}_{n}={u}_{n}\in cone(clA). So, for each n, there exist {t}_{n}>0, {a}_{n}\in clA such that
On the one hand, if {a}_{n}\to {0}_{X}, then {t}_{n}\to +\mathrm{\infty} as n\to +\mathrm{\infty}. For each n, we have
For given \epsilon >0, there exists {n}_{0} such that {a}_{n}\in B({0}_{X},\epsilon ), \mathrm{\forall}n\ge {n}_{0}. We divide it into two cases to discuss.

(i)
If \{{a}_{n}\}\subset clA\setminus intA as n\ge {n}_{0}, by assumption, we can conclude that there exists a subsequence \{{a}_{{n}_{m}}\}\subseteq \{{a}_{n}\} such that
{a}_{{n}_{m}}={\lambda}_{{n}_{m}}{a}_{{\overline{n}}_{0}},(2.5)
with {\lambda}_{{n}_{m}}\in (0,1] and {\overline{n}}_{0}\ge {n}_{0}. In fact, since {t}_{n}{a}_{n}\to u, so for any {\epsilon}_{1}>0, there exists {n}_{1}\ge {n}_{0} such that \parallel {t}_{n}{a}_{n}u\parallel \le {\epsilon}_{1} as n\ge {n}_{1}. If for any given {\overline{n}}_{0}\ge {n}_{1} and \lambda \in (0,1],
then we can take some {t}_{n} from \{{t}_{n}\} such that \frac{{t}_{{\overline{n}}_{0}}}{{t}_{n}}<1 as n\ge {\overline{n}}_{0}. From (2.6) it follows that \frac{{t}_{{\overline{n}}_{0}}}{{t}_{n}}{a}_{{\overline{n}}_{0}}\notin \{{a}_{n}\} as n\ge {\overline{n}}_{0}. Thus \parallel {t}_{n}(\frac{{t}_{{\overline{n}}_{0}}}{{t}_{n}}{a}_{{\overline{n}}_{0}})u\parallel =\parallel {t}_{{\overline{n}}_{0}}{a}_{{\overline{n}}_{0}}u\parallel >{\epsilon}_{1}, a contradiction. So (2.5) holds. Again from {t}_{n}{a}_{n}\to u, so, {t}_{{n}_{m}}{a}_{{n}_{m}}={t}_{{n}_{m}}{\lambda}_{{n}_{m}}{a}_{{\overline{n}}_{0}}\to u. Furthermore, {a}_{{\overline{n}}_{0}} is fixed, so {t}_{{n}_{m}}{\lambda}_{{n}_{m}}\to {t}_{0}. That is, {t}_{{n}_{m}}{\lambda}_{{n}_{m}}={t}_{0}+{\overline{t}}_{{n}_{m}} with {\overline{t}}_{{n}_{m}}\to 0. Thus, {t}_{{n}_{m}}{a}_{{n}_{m}}=({t}_{0}+{\overline{t}}_{{n}_{m}}){a}_{{\overline{n}}_{0}}={t}_{0}{a}_{{\overline{n}}_{0}}+{\overline{t}}_{{n}_{m}}{a}_{{\overline{n}}_{0}}\to {t}_{0}{a}_{{\overline{n}}_{0}}. From the uniqueness of limits, we have u={t}_{0}{a}_{{\overline{n}}_{0}} with {a}_{{\overline{n}}_{0}}\in clA\setminus intA, so u\in cone(clA).

(ii)
If \{{a}_{n}\}\subset intA, then for any n, \mathrm{\exists}{\epsilon}_{n} such that B({a}_{n},{\epsilon}_{n})\subset A. From {t}_{n}{a}_{n}=u+{x}_{n}, one has {a}_{n}\frac{u}{{t}_{n}}=\frac{{x}_{n}}{{t}_{n}} and {a}_{n}\in B({0}_{X},\epsilon ) as n is large enough. If there exists {n}_{0} such that \parallel {a}_{{n}_{0}}\frac{u}{{t}_{{n}_{0}}}\parallel =\frac{\parallel {x}_{{n}_{0}}\parallel}{{t}_{{n}_{0}}}\le {\epsilon}_{{n}_{0}}, then \frac{u}{{t}_{{n}_{0}}}\in A, so u\in cone(clA). If, for any n, \parallel {a}_{n}\frac{u}{{t}_{n}}\parallel =\parallel \frac{{x}_{n}}{{t}_{n}}\parallel >{\epsilon}_{n}, then this implies there exist {\overline{a}}_{n}\in clA\setminus A and \parallel {\overline{a}}_{n}{a}_{n}\parallel \le \frac{\parallel {x}_{n}\parallel}{{t}_{n}}. However, \parallel {t}_{n}{\overline{a}}_{n}u\parallel =\parallel {t}_{n}{\overline{a}}_{n}{t}_{n}{a}_{n}+{t}_{n}{a}_{n}u\parallel \le {t}_{n}\frac{\parallel {x}_{n}\parallel}{{t}_{n}}+\parallel {x}_{n}\parallel =2\parallel {x}_{n}\parallel \to 0. Thus, {t}_{n}{\overline{a}}_{n}\to u. By {\overline{a}}_{n}\in clA\setminus A and (i), we conclude that u\in cone(clA).
On the other hand, if {a}_{n}\nrightarrow {0}_{X}, then \{{t}_{n}\} is bounded. Set t={sup}_{n}\{{t}_{n}\}, then t\ge {t}_{n} and t\in {R}^{+}. Dividing t by (2.4) and for each n\in N, set {b}_{n}=\frac{{t}_{n}{a}_{n}}{t}=\frac{u}{t}+\frac{{x}_{n}}{t}. According to assumption A satisfying the property Ϝ, we get {b}_{n}\in A and {b}_{n}\to \frac{u}{t} as n\to \mathrm{\infty}. Thus, \frac{u}{t}\in clA, i.e., u\in cone(clA). From the above two parts, we get cone(clA) is a closed cone, so cl(coneA)\subseteq cone(clA). The proof is complete. □
Corollary 2 Let A be a cone of X and satisfy the property Λ near the {0}_{X}, then cone(clA)=clA.
Corollary 3 Let {x}_{0}\in A, if A is a closed convex subset of X such that A{x}_{0} satisfy the property Λ near the {0}_{X}, then cone(A{x}_{0})=clcone(A{x}_{0}).
Proposition 2.4 (see Chapter 4 of [3])
Let {x}_{0}\in A\subset X, if A is starshaped at {x}_{0}, then T(A,{x}_{0})=cl(cone(A{x}_{0})).
Corollary 4 Let {x}_{0}\in A\subset X, if A is a closed convex subset such that A{x}_{0} satisfy the property Λ near the {0}_{X}, then T(A,{x}_{0})=cone(A{x}_{0}).
Proof Since {x}_{0}\in A and A is a convex subset, we have T(A,{x}_{0})=cl(cone(A{x}_{0})). To apply Corollary 3, one has
it is clear that T(A,{x}_{0})=cone(A{x}_{0}). □
3 The existence theory for contingent epiderivative
The aim of this section is to discuss the existence of the contingent epiderivative for a setvalued map defined from a real normed space to a real normed space. Let A\subset X, F:A\to {2}^{Y} be a setvalued mapping.
Definition 3.1 (see [1])
Let {x}_{0}\in A and a pair ({x}_{0},{y}_{0})\in grF be given. The contingent epiderivative DF({x}_{0},{y}_{0}) of F at ({x}_{0},{y}_{0}) is the singlevalued mapping from X to Y defined by
Next, we establish DF({x}_{0},{y}_{0}) in the following (3.6) and also prove it is a singlevalued mapping.
Let a pair ({x}_{0},{y}_{0})\in grF, for any x\in X, set
Notice that if A\subset X is a closed convex set, F is Dconvex and epiF is a closed subset in X\times Y such that (epiF({x}_{0},{y}_{0})) satisfies the property Λ near the ({0}_{X},{0}_{Y}), then
is a closed convex cone. So, for every (x,y)\in epi(DF({x}_{0},{y}_{0})) and (x,y)\ne ({0}_{X},{0}_{Y}), if (x,y)\in T(epiF,({x}_{0},{y}_{0})), then there exist l>0, a\in A, v\in F(a) and d\in D such that
By (3.3), we have
and
where \lambda =\frac{1}{l}.
Otherwise, if (x,y)=({0}_{X},{0}_{Y}), then we can conclude that a={x}_{0}, v={y}_{0} and d={y}_{0}. Therefore, formulas (3.4) and (3.5) are clearly established.
Remark 3.1 From the above discussion and Corollary 4, if A\subset X is a closed convex set and A{x}_{0} satisfies the property Λ near the {0}_{X}, F:A\to {2}^{Y} is Dconvex and epiF is closed in X\times Y such that (epiF({x}_{0},{y}_{0})) satisfies the property Λ near the ({0}_{X},{0}_{Y}), then DF({x}_{0},{y}_{0}) can be defined on T(A,{x}_{0}).
Furthermore, we can conclude that formula (3.2) is equivalent to the following:
where x\in T(A,{x}_{0}) and \lambda \ge 0.
From Theorem 2.1, we can obtain the following existence theorem.
Theorem 3.1 Let A\subset X be a closed convex set, let F:A\to {2}^{Y} be Dconvex, and let epiF be a closed subset in X\times Y. Let ({x}_{0},{y}_{0})\in grF. If for any x\in T(A,{x}_{0}) and \lambda \ge 0, the following conditions hold: (1) (F({x}_{0}+\lambda x){y}_{0}+D) satisfies the properties Ϝ and Λ near the {0}_{Y}; (2) cone(F({x}_{0}+\lambda x){y}_{0}+D) is Dlower bounded; (3) the cone D is Daniell cone; (4) cone(F({x}_{0}+\lambda x){y}_{0}+D) satisfies (2.1). Then IMin(cone(F({x}_{0}+\lambda x){y}_{0}+D))\ne \mathrm{\varnothing} is a singlepoint set.
Proof Since epiF a closed convex subset in X\times Y, by Proposition 2.1, one has F({x}_{0}+\lambda x)+D a closed set for each x\in T(A,{x}_{0}), so is (F({x}_{0}+\lambda x){y}_{0}+D). By assumption (1) and Proposition 2.3, we can conclude that cone(F({x}_{0}+\lambda x){y}_{0}+D) is a closed set. Considering Theorem 2.1 and given conditions (2) and (3), we have Min(cone(F({x}_{0}+\lambda x){y}_{0}+D))\ne \mathrm{\varnothing}. Combining Remark 2.1 and assumption (4), we can conclude IMin(cone(F({x}_{0}+\lambda x){y}_{0}+D))\ne \mathrm{\varnothing} is a singlepoint set. □
Remark 3.2 From (3.6), we know that DF({x}_{0},{y}_{0}) exists if IMin(cone(F({x}_{0}+\lambda x){y}_{0}+D))\ne \mathrm{\varnothing}.
The following theorem shows that our definition of the contingent epiderivative for a setvalued mapping is well defined.
Theorem 3.2 If all the conditions of Theorem 3.1 are fulfilled, then epi(DF({x}_{0},{y}_{0}))=T(epiF,({x}_{0},{y}_{0})).
Proof Combining (3.6) and Theorem 3.1, we get DF({x}_{0},{y}_{0})(x) exists for each x\in T(A,{x}_{0}). Since (x,DF({x}_{0},{y}_{0})(x))\in cone(epiF({x}_{0},{y}_{0})), thus
Now, let us show the converse inclusion. From the definition of the DF({x}_{0},{y}_{0}), for any (x,y)\in T(epiF,({x}_{0},{y}_{0})), we have
Therefore,
The proof is complete. □
Next we show under appropriate assumptions that the contingent epiderivative is a strictly positive homogeneous and subadditive map in the case of A, a convex set, and epiF, a closed convex subset.
Definition 3.2 Let X be a real linear space and let Y be a real linear space partially ordered by a closed convex pointed cone D\subset Y. A map f:X\to {2}^{Y} is called

(a)
strictly positive homogeneous if
f(\alpha x)=\alpha f(x)\phantom{\rule{1em}{0ex}}\text{for all}\alpha 0\text{and all}x\in X, 
(b)
subadditive if
f({x}_{1})+f({x}_{2})\subset f({x}_{1}+{x}_{2})+D\phantom{\rule{1em}{0ex}}\text{for all}{x}_{1},{x}_{2}\in X.
If the properties under (a) with \alpha \ge 0 and (b) hold, then f is called sublinear.
Theorem 3.3 Let A be a closed convex set in X, {x}_{0}\in A such that A{x}_{0} satisfies the property Λ near the {0}_{X}. Let D be a closed convex pointed cone in Y and ({x}_{0},{y}_{0})\in grF such that epiF is a closed convex set and (epiF({x}_{0},{y}_{0})) satisfies the property Λ near the ({0}_{X},{0}_{Y}). If for all x\in T(A,{x}_{0}), DF({x}_{0},{y}_{0})(x) exists, then DF({x}_{0},{y}_{0})(x) is sublinear.
Proof We take any \alpha >0 and any x\in T(A,{x}_{0}). Then we obtain
Thus DF({x}_{0},{y}_{0}) is strictly positive homogeneous.
Next, for {x}_{1},{x}_{2}\in T(A,{x}_{0}), we have ({x}_{1},DF({x}_{0},{y}_{0})({x}_{1}))\in T(epiF,({x}_{0},{y}_{0})) and ({x}_{2},DF({x}_{0},{y}_{0})({x}_{2}))\in T(epiF,({x}_{0},{y}_{0})). Since F is Dconvex and epiF is a closed subset, then T(epiF,({x}_{0},{y}_{0})) is a closed convex cone. Thus,
implies
By Remark 2.1, we have
The proof is complete. □
4 Relationships between vector variationallike inequalities and optimization problems
In this section, we restrict ourselves to dealing with relationships between two kinds of vector variationallike inequality problems and setvalued optimization problems.
The setvalued vector optimization problem under our consideration is
where F:X\to {2}^{Y}, F(A)={\bigcup}_{x\in A}F(x). We denote setvalued optimization problem (4.1) as (SOP).
Definition 4.1 Consider the above problem (SOP), let {x}_{0}\in A, {y}_{0}\in F({x}_{0}).

(i)
A pair ({x}_{0},{y}_{0})\in grF is called a minimal solution of F on A if (F(A){y}_{0})\cap D=\{{0}_{Y}\};

(ii)
A pair ({x}_{0},{y}_{0})\in grF is called an ideal minimal solution of F on A if (F(A){y}_{0})\subseteq D.
The sets of all minimal solutions and ideal minimal solutions of (SOP) are denoted by Min(F,A) and IMin(F,A), respectively.
In the following, we always assume that contingent epiderivative of F exists.
Definition 4.2 (see [14])
Let the set A be convex, let the setvalued mapping F:A\to {2}^{Y} be Dconvex. Let ({x}_{0},{y}_{0})\in grF and let the contingent epiderivative DF({x}_{0},{y}_{0}) of F at ({x}_{0},{y}_{0}) exist.

(i)
A linear map L:X\to Y, with L(x)\le DF({x}_{0},{y}_{0})(x), for all x\in T(A,{x}_{0}) is called a subgradient of F at ({x}_{0},{y}_{0}).

(ii)
The set \partial F({x}_{0},{y}_{0})=\{L:X\to Y:L(x)\le DF({x}_{0},{y}_{0})(x),\mathrm{\forall}x\in T(A,{x}_{0})\} of all subgradients L of F at ({x}_{0},{y}_{0}) is called the subdifferential of F at ({x}_{0},{y}_{0}).
Definition 4.3 A set A\subset X is said to be an invex set if there exists a function \eta :X\times X\to X such that \mathrm{\forall}x,y\in A, \mathrm{\forall}\lambda \in [0,1], y+\lambda \eta (x,y)\in A.
Throughout the paper, we always assume that A is an invex subset of X, the function η defined on A, i.e., \eta :A\times A\to X and the subdifferential of F exists at every (x,y)\in grF.
Definition 4.4 Let F:A\to {2}^{Y} be a setvalued mapping, ({x}_{0},{y}_{0})\in grF. F is called strong pseudoinvex at ({x}_{0},{y}_{0}) with respect to η on A if \mathrm{\forall}x\in A, \mathrm{\forall}y\in F(x) and \mathrm{\forall}L\in \partial F({x}_{0},{y}_{0}), y{y}_{0}\notin D\Rightarrow L\eta (x,{x}_{0})\notin D.
F is said to be strong pseudoinvex with respect to η on A if for every pair (x,y)\in grF, F is strong pseudoinvex at (x,y) with respect to η.
Definition 4.5 Let F:A\to {2}^{Y} be a setvalued mapping, ({x}_{0},{y}_{0})\in grF. F is called pseudoinvex at ({x}_{0},{y}_{0}) with respect to η on A if \mathrm{\forall}x\in A, \mathrm{\forall}y\in F(x) and \mathrm{\forall}L\in \partial F({x}_{0},{y}_{0}), y{y}_{0}\in D\Rightarrow L\eta (x,{x}_{0})\in D.
F is said to be pseudoinvex with respect to η on A if for every pair (x,y)\in grF, F is pseudoinvex at (x,y) with respect to η.
Definition 4.6 Let F:A\to {2}^{Y} be a setvalued mapping. F is said to be Dpreinvex with respect to η on A if \mathrm{\forall}x,y\in A, \mathrm{\forall}\lambda \in [0,1], \lambda F(x)+(1\lambda )F(y)\subset F(y+\lambda \eta (x,y))+D.
From the above definitions, we have the following useful proposition.
Proposition 4.1 Let F:A\to {2}^{Y} be Dpreinvex with respect to η on A, ({x}_{0},{y}_{0})\in grF. If DF({x}_{0},{y}_{0}) exists on cone(A{x}_{0}), then for all L\in \partial F({x}_{0},{y}_{0}), all x\in A, all y\in F(x), y{y}_{0}\in \u3008L,\eta (x,{x}_{0})\u3009+D.
Proof For all x\in A, since F is Dpreinvex with respect to η on A and {x}_{0}\in A, for any \lambda \in [0,1], we obtain
which implies that
Thus, we get
It is clear that
By the definition of subdifferential \partial F({x}_{0},{y}_{0}), for all L\in \partial F({x}_{0},{y}_{0}), we have
that is,
Combining (4.3), (4.4) and (4.5), we can conclude, y{y}_{0}\in \u3008L,\eta (x,{x}_{0})\u3009+D. □
Remark 4.1 It is easy to see that F is pseudoinvex with respect to η on A if F is Dpreinvex with respect to η on A.
A vector variational inequality has been shown to be a useful tool in vector optimization. Some authors have proved the equivalence between them, see [9, 10, 15]. The vector variationallike inequality problem is a generalized form of the vector variational inequality problem, which was introduced and studied by [8].
Now, let us offer the following definitions.
A vector variationallike inequality problem (VVLI) is to find ({x}_{0},{y}_{0})\in grF such that \u3008L,\eta (x,{x}_{0})\u3009\notin D\setminus \{{0}_{Y}\} for all x\in A\setminus \{{x}_{0}\} and all L\in \partial F({x}_{0},{y}_{0}). A strong vector variationallike inequality problem (SVVLI) is to find ({x}_{0},{y}_{0})\in grF such that \u3008L,\eta (x,{x}_{0})\u3009\in D for all x\in A and all L\in \partial F({x}_{0},{y}_{0}).
In the following, using the tools of nonsmooth analysis and the concept of nonsmooth (strong) vector pseudoinvexity, we shall obtain the stronger results than those of [11].
Theorem 4.1 Let {x}_{0}\in A, ({x}_{0},{y}_{0})\in grF. If the pair ({x}_{0},{y}_{0}) solves the (SVVIL) and F is strong pseudoinvex at ({x}_{0},{y}_{0}) with respect to η on A, then ({x}_{0},{y}_{0}) is an ideal minimizer of the (SOP).
Proof Since the pair ({x}_{0},{y}_{0}) solves the (SVVIL), we have that
If ({x}_{0},{y}_{0}) is not an ideal minimizer of the (SOP), then there exist \overline{x}\in A, (\overline{x},\overline{y})\in grF such that
Since F is strong pseudoinvex at ({x}_{0},{y}_{0}) with respect to η on A, we get \overline{x}\in A, \mathrm{\forall}L\in \partial F({x}_{0},{y}_{0}) such that
By (4.7) and (4.8), we obtain \u3008L,\eta (\overline{x},{x}_{0})\u3009\notin D, which contradicts (4.6). □
In order to see the converse of the above theorem, we must impose stronger conditions, as can be observed in the following theorem.
Theorem 4.2 Suppose that F:A\to {2}^{Y} and −F is Dpreinvex with respect to η on A, ({x}_{0},{y}_{0})\in grF. If \partial F({x}_{0},{y}_{0})\subseteq \partial (F)({x}_{0},{y}_{0}) and the pair ({x}_{0},{y}_{0}) is an ideal minimizer of (SOP), then ({x}_{0},{y}_{0}) solves the (SVVLI).
Proof Let ({x}_{0},{y}_{0}) be an ideal minimizer of the (SOP), then we have
Since −F is Dpreinvex with respect to η and \partial F({x}_{0},{y}_{0})\subseteq \partial (F)({x}_{0},{y}_{0}), we have that \mathrm{\forall}x\in A, \mathrm{\forall}y\in F(x) and \mathrm{\forall}L\in \partial F({x}_{0},{y}_{0}) such that
which implies
Combining (4.9) and (4.10), we can conclude
for all x\in A and all L\in \partial F({x}_{0},{y}_{0}). That is, ({x}_{0},{y}_{0}) solves the (SVVLI). □
Theorem 4.3 Suppose that F:A\to {2}^{Y} and −F is Dpreinvex with respect to η on A, ({x}_{0},{y}_{0})\in grF. If \partial F({x}_{0},{y}_{0})\subseteq \partial (F)({x}_{0},{y}_{0}) and the pair ({x}_{0},{y}_{0}) is a minimizer of (SOP), then ({x}_{0},{y}_{0}) solves the (VVLI).
Proof Let ({x}_{0},{y}_{0}) be a minimizer of the (SOP), then we have
If ({x}_{0},{y}_{0}) does not solve the (VVIL), then there exist \overline{x}\in A\setminus \{{x}_{0}\} and \overline{L}\in \partial F({x}_{0},{y}_{0}) such that
Since −F is Dpreinvex with respect to η and \partial F({x}_{0},{y}_{0})\subseteq \partial (F)({x}_{0},{y}_{0}), we have that \mathrm{\forall}x\in A, \mathrm{\forall}y\in F(x) and \overline{L}\in \partial F({x}_{0},{y}_{0}) such that
Combining (4.12) and (4.13), for all \overline{y}\in F(\overline{x}), one has
which contradicts (4.11). The proof is complete. □
The converse case of the above theorem (see Theorem 4.4 of [11]) only requires F to be pseudoinvex.
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This work was supported by the Fundamental Research Funds for the Central Universities, No. K5051370004.
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YC carried out the main preliminaries, the existence theory for contingent epiderivative and the relationships between vector variationallike inequalities and optimization problems. SL carried out the introduction and some preliminaries. YL participated in the design of the study and some preliminaries.
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Chai, Y., Liu, S. & Li, Y. The existence of contingent epiderivative for a setvalued mapping and vector variationallike inequalities. J Inequal Appl 2013, 352 (2013). https://doi.org/10.1186/1029242X2013352
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DOI: https://doi.org/10.1186/1029242X2013352