On local quasi efficient solutions for nonsmooth vector optimization

We are interested in local quasi efficient solutions for nonsmooth vector optimization problems under new generalized approximate invexity assumptions. We formulate necessary and sufficient optimality conditions based on Stampacchia and Minty types of vector variational inequalities involving Clarke's generalized Jacobians. We also establish the relationship between local quasi weak efficient solutions and vector critical points

Fuzzy games with a countable space of actions and applications to systems of generalized quasi-variational inequalities

In this paper, we introduce an abstract fuzzy economy (generalized fuzzy game) model with a countable space of actions and we study the existence of the fuzzy equilibrium. As applications, two types of results are obtained. The first ones concern the existence of the solutions for systems of generalized quasi-variational inequalities with random fuzzy mappings which we define. The last ones are new random fixed point theorems for correspondences with values in complete countable metric spaces.Comment: 18 page

Existence of solutions and star-shapedness in Minty variational inequalities

Minty variational inequalities have proven to define a stronger notion of equilibrium than Stampacchia variational inequalities. This conclusion leads to argue that some regularity, e.g. convexity or generalized convexity, has to be implicit for any function that admits a solution of the corresponding integrable Minty variational inequality. Quasi-convexity arises almost naturally when functions of one variable are involved. However some differences appear when considering functions of several variables. In this case we show that existence of a solution does not necessarily imply quasi-convexity of the function and instead we prove that the level sets of the function must be star-shaped at a point which is a solution of the Minty variational inequality.Minty variational inequality, generalized convexity, star-shaped sets, existence of solutions

Generalized Games and Non-compact Quasi-variational Inequalities

AbstractIn this paper, by developing an approximation approach which is originally due to Tuleca in 1986, we prove the existence of equilibria for generalized games in which constraint mappings (correspondences) are lower (resp., upper) semicontinuous instead of having lower (resp., upper) open sections or open graphs in infinite dimensional topological spaces. Then, existence theorems of solutions for quasi-variational inequalities and non-compact generalized quasi-variational inequalities are also established. Finally, existence theorems of constrained games with non-compact strategy sets are derived. Our results unify and generalize many well known results given in the existing literature. In particular, we answer the question raised by Yannelis and Prabhakar in 1983 in the affirmative under more weaker conditions

Generalized bi-quasi-variational inequalities

AbstractLet E, F be Hausdorff topological vector spaces over the field Φ (which is either the real field or the complex field), let 〈 , 〉: F × E → Φ be a bilinear functional, and let X be a non-empty subset of E. Given a multi-valued map S: X → 2x and two multi-valued maps M, T: X → 2F, the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point ŷ ϵ X such that ŷ ϵ S(ŷ) and infw ϵ T(ȳ)Re〈ƒ − w, ŷ − x〉 ⩽ 0 for all x ϵ S(ŷ) and for all ƒ ϵ M(ŷ). In this paper two general existence theorems on solutions of GBQVIs are obtained which simultaneously unify, sharpen, and extend existence theorems for multi-valued versions of Hartman-Stampacchia variational inequalities proved by Browder and by Shih and Tan, variational inequalities due to Browder, existence theorems for generalized quasi-variational inequalities achieved by Shih and Tan, theorems for monotone operators obtained by Debrunner and Flor, Fan, and Browder, and the Fan-Glicksberg fixed-point theorem

Charactarizations of Linear Suboptimality for Mathematical Programs with Equilibrium Constraints

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasi variational inequalities, implicit complementarity problems, etc

Sufficient optimality criteria and duality for multiobjective variational control problems with G-type I objective and constraint functions

In the paper, we introduce the concepts of G-type I and generalized G-type I functions for a new class of nonconvex multiobjective variational control problems. For such nonconvex vector optimization problems, we prove sufficient optimality conditions for weakly efficiency, efficiency and properly efficiency under assumptions that the functions constituting them are G-type I and/or generalized G-type I objective and constraint functions. Further, for the considered multiobjective variational control problem, its dual multiobjective variational control problem is given and several duality results are established under (generalized) G-type I objective and constraint functions