Vector variational inequalities and their relations with vector optimization

In this paper, K c quasiconvex, K c pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where K is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) K -pseudomonotonicity, (strict) K -naturally quasimonotonicity and K -quasimonotonicity of Clarke subdifferential maps have also been defined. Further, we introduce Minty weak (MVVIP) and Stampacchia weak (SVVIP) vector variational inequalities over arbitrary cones. Under regularity assumption, we have proved that a weak minimum solution of vector optimization problem (VOP) is a solution of (SVVIP) and under the condition of K c pseudoconvexity we have obtained the converse for MVVIP (SVVIP). In the end we study the interrelations between these with the help of strict K -naturally quasimonotonicity of Clarke subdifferential map.


Introduction
The Variational inequality problem was first introduced by Hartman and Stampacchia [1] in their seminal paper.As it has many applications in fundamental sciences as well as in economics and management sciences, it has become very popular among researchers.Giannessi [2] introduced vector variational inequality problem and since then a great deal of research started in this area by various authors like Chen and Yang [3], Chen [4], Yao [5], Giannessi [6,7,8], Komlosi [9], Yang et al. [10], Lee and Lee [11] etc. Giannessi [6] has shown equivalence between efficient solutions of differentiable, convex vector optimization problem and solutions of variational inequality of Minty type.Yang et al. [10] established some relations between a solution of a Minty vector variational inequality (VVI) problem and an efficient solution of a differentiable nonconvex vector optimization problem under the assumptions of pseudoconvexity of functions or pseudomonotonicity of their gradients.Komlosi [9] studied Stampacchia and Minty vector variational inequality problems and discussed the solution concepts of these problems and vector optimization problem.Vector variational-like inequality problems have been studied by many authors like Mishra and Wang [12], Chinaie et al. [13] etc. Recently, Rezaie and Zafarani [14] have studied relations between vector optimization problem and Minty and Stampacchia vector variational-like inequalities over cones contained in n + R \ {0} for nondifferentiable functions under generalized invexity or generalized monotonicity assumptions.
Generalized monotonicity plays a central role in the study of the existence of solution of variational inequality problems and their relations with vector optimization problems.Monotonicity (generalized) concepts have been related to convexity (generalized) of functions in case of gradient maps by various authors like Karamardian and Schaible [15], Hadjisavvas and Schaible [16] and Schaible [17].Various authors like Cambini [18], Cambini and Martein [19], Vani [20] etc. have extended generalized convexity and / or generalized monotonicity concepts from scalar case to the vector valued functions.In this paper, we introduce the notions of Kc quasiconvex, weakly K -c quasiconvex, K -c pseudoconvex and strict K -c pseudoconvex functions in terms of Clarke subdifferentials.The interrelations between above mentioned functions have been given.The concepts of (strict, weakly) Kpseudomonotonicity, (strict) K -naturally quasimonotonicity and K -quasimonotonicity of Clarke subdifferential maps have been defined.Further Minty weak and Stampacchia weak vector variational inequalities over arbitrary cones have been introduced.Their relations with vector optimization problem over cones have been studied with the help of K -c pseudoconvex functions.We end the paper by presenting interrelations among Minty weak and Stampacchia weak vector variational inequalities over cones under the assumption of strict K -naturally quasimonotonicity of Clarke subdifferential map.

Generalized Nonsmooth Cone Convexity and Generalized Cone Monotonicity
We begin this section with the following definitions.

Definition 1[21]
Let : R n R be a locally Lipschitz function on R n .Then the Clarke generalized subdifferential of at x R n is given as Let K R k be a closed convex, pointed cone with non empty interior and let int K and K denote the interior and closure of K respectively.The positive dual cone K (Swaragi, Nakayama and Tanino [23]) is defined as R n be a nonempty subset.Then the convex hull of A is denoted by .co A Jahn [24] defined K -quasiconvex function as given below.

Definition 2. The function
We now introduce two important classes of generalized convex functions (with respect to cones) using the concept of Clarke subdifferential namely K -c quasiconvex and weakly K -c quasiconvex functions.
Following definition has been introduced on the lines of Cambini [18] and Jahn [24].
then the above definition reduces to the definition of cquasiconvex function given by Bector, Chandra and Dutta [22].
(ii) If k=1, K R + and f is continuously differentiable then the above definition reduces to the definition of quasiconvex function.
On the lines of Cambini [18], we give below the definition of weakly K -c quasiconvex function.

Definition 4. The function
We now introduce the following definition of Kc pseudoconvex and strict K -c pseudoconvex functions on the lines of Cambini [18].
then the above definition reduces to the definition of cpseudoconvex function given by Bector, Chandra and Dutta [22].
(ii) If f is continuously differentiable function then the above definition reduces to the definition of K -pseudoconvex function given by Aggarwal [25] and Cambini and Martein [19] as We now present interrelations between the above defined functions in the form of following remarks.
Remark 5 Every K -c quasiconvex function is weakly K -c quasiconvex function.
On the lines of Rezaie and Zafarani [14], we now give the definitions of generalized monotone set valued maps over arbitrary closed convex and pointed cones with non empty interior.2 , 0 ,0 ( ) , ( ) ,0 ,0 The section proceeds further by presenting interrelationships between above mentioned generalized monotone maps.
Remark 13 Every strict K -pseudomonotone map is K -pseudomonotone but converse may not be true as can be seen from the following example: Every strict K -naturally quasimonotone map is K -naturally quasimonotone but converse may not be true as can be seen from the following example: ,0 ,0 ( ) , ( ) 0, 0 ,0

Vector Variational Inequalities
In this section we consider the following vector optimization problem over cones and study its relation with associated vector variational inequalities: Then by Theorem 5.4.1 of Bector, Chandra and Dutta[22], f is quasiconvex function.Thus it characterizes Lipschitz quasiconvex function.We now provide an example of K -c quasiconvex function.
We now give below an example of K -c pseudoconvex function.

Definition 7 .Definition 8 .Definition 9 .Definition 10 .Definition 11 .Definition 12 .
The set valued map c f is Kpseudomonotone on D if for every pair of distinct points , definition becomes the definition of K -pseudomonotone map given by Rezaie and Zafarani[14].(ii)If k=1, K R + and f is continuously differentiable then the above definition reduces to the following definition of pseudomonotonicity of the map c ff (Karamardian and Schaible [The set valued map c f is strict K -pseudomonotone on D if for every pair of distinct points , definition becomes the definition of strict K -pseudomonotone map given by Rezaie and Zafarani[14].(ii)If k=1, K R + and f is continuously differentiable then the above definition reduces to the The set valued map c f is weakly K -pseudomonotone on D if for distinct points , The set valued map c f is Knaturally quasimonotone on D if for every pair of distinct points , definition becomes the definition of K -quasimonotone map given by Rezaie and Zafarani[14].(ii)If k=1, K R + and f is continuously differentiable then the above definition reduces to the following definition of quasimonotonicity of the map c ff (Karamardian and Schaible [The set valued map c f is strict K -naturally quasimonotone on D if for every pair of distinct points , definition becomes the definition of strict K -quasimonotone map given by Rezaie and Zafarani[14].(ii)If k=1, K R + and f is continuously differentiable then the above definition reduces to pseudomonotonicity of the map c ff (Karamardian and Schaible[15]) The set valued map c f is Kquasimonotone on D if for every pair of distinct points , If k=1, K R + and f is continuously differentiable then the above definition reduces to pseudomonotonicity of the map c ff (Karamardian and Schaible[15]) ( ), 0 f x x y ( ), 0. f y x y Now we give an example of strict K -pseudomonotone map as follows: + and f is continuously differentiable then the above definition reduces to the definition of pseudoconvex function.
ik is locally Lipschitz, K is closed convex and pointed cone with nonempty interior in R k and C is convex subset of .If xC is a solution of (MVVIP) and f is K -c pseudoconvex then x is a weak minimum for (VOP).Proof Let xC be a solution of (MVVIP).Suppose on contrary x is not a solution of (SVVIP).Then there exist yC such that At present she is supervising 2 M.Phil.and4Ph.D. students.Her areas of interest include generalized convexity, vector optimization, nonsmooth optimization, set valued optimization and variational inequality problems.Bhawna Kohli completed her M.Sc. in 2001 and M.Phil. in 2004 from the Department of Mathematics, University of Delhi.At present she has submitted her Ph.D. thesis in the area of Mathematical programming in the Department of Mathematics, University of Delhi.She has published 5 research papers in various well known journals.Her areas of interest include Nonsmooth Optimization, vector optimization, bilevel programming and variational inequality problems.
D Definition 13.A vector xC is said to be a weak minimum of (VOP) if c fzwe have || || ' .k( ),