ON GENERALIZED VECTOR QUASI-VARIATIONAL TYPE INEQUALITIES

Some generalized vector variational inequalities and complementarity problems for multivalued mappings

In a recent paper, Domokos and Kolumban introduced variational inequalities with operator solutions to provide a unified approach to several kinds of variational inequalities and vector variational inequalities in Banach spaces. Inspired by their work, in a former paper, we extended the scheme of vector variational inequalities with operator solutions from the single-valued case to the multivalued one and provided some applications to generalized vector variational inequalities and generalized quasivector variational inequalities in normed spaces. As a continuation of the former work, in this paper, we further extend those results to more general and tangible cases in the context of Hausdorff topological vector spaces or locally convex spaces.

In this paper, we consider vector variational inequalities with set-valued mappings over countable product sets in a real Banach space setting. By employing concepts of relative pseudomonotonicity, we establish several existence results for generalized vector variational inequalities and for systems of generalized vector variational inequalities. These results strengthen previous existence results which were based on the usual monotonicity type assumptions

This paper is concerned with gap functions of generalized vector variational inequalities (GVVI). By using scalarization approach, scalar-valued variational inequalities of (GVVI) are introduced. Some relationships between the solutions of (GVVI) and its scalarized versions are established. Then, by using these relationships and some mild conditions, scalar-valued gap functions for (GVVI) are established.

In this paper, we give notions of well posedness for a vector optimization problem and a vector variational inequality of the differential type. First, the basic properties of well-posed vector optimization problems are studied and the case of C-quasiconvex problems is explored. Further, we investigate the links between the well posedness of a vector optimization problem and of a vector variational inequality. We show that, under the convexity of the objective function f, the two notions coincide. These results extend properties which are well known in scalar optimization.

Over the past several years the f inite-dimensional variational inequality problem has been well studied from the perspectives of computation, sensitivity analysis and application. This paper will briefly review the variational inequality literature and will also discuss the finite dimensional quasivariational inequality problem from the above mentioned perspectives. In particular, three recent results will be discussed in detail. First, an acceleration step for the nonlinear Jacobi and projection algorithms will be presented along with empirical results. Second, a special case of the quasivariational inequality problem which arises in pseudo-Nash or social equilibria games is studied in detail. Characterizations of the solutions of such problems and their relationships with variational inequality solution are presented along with some results on the stability and sensitivity of these solutions. Finally, the theory and empirical validation of a restricted simplicial decomposition algorithm for large-scale, linearly-constrained variational inequalities will be described.

This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem with the uniqueness assumption of the multipliers replaced by a convexity assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed linear complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric variational inequality, the use of the generalized linear complementarity problem as a tool to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affine variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and various stability theorems under some weak second-order conditions.

In this paper, K- quasiconvex, K- pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) -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- 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.

In this paper, we employ the characterization for an approximate convex function in terms of its convexificator to establish the relationships between the solutions of Stampacchia type vector variational inequality problems in terms of convexificator and quasi efficient solution of a nonsmooth vector optimization problems involving locally Lipschitz functions. We identify the vector critical points, the weak quasi efficient points and the solutions of the weak vector variational inequality problem under generalized approximate convexity assumptions. The results of the paper extend, unify and sharpen corresponding results in the literature. In particular, this work extends and generalizes earlier works by Giannessi [11], Upadhyay et al. [31], Osuna-Gomez et al. [30], to a wider class of functions, namely the nonsmooth approximate convex functions and its generalizations. Moreover, this work sharpens earlier work by Daniilidis and Georgiev [5] and Mishra and Upadhyay [23], to a more general class of subdifferentials known as convexificators.

This paper deals with the Quasi Variational Inequality (QVI) problem on Banach spaces.Necessary and sufficient conditions for the solutions of QVI are given, using the subdifferential of distance function and the normal cone.A dual problem corresponding to QVI is constructed and strong duality is established.The solutions of dual problem are characterized according to the saddle points of the Lagrangian map.A gap function for dual of QVI is presented and its properties are established.Moreover, some applied examples are addressed.

<abstract><p>This paper establishes connections between the solutions of some new vector controlled variational inequalities and (proper) efficient solutions of the corresponding multiobjective controlled variational problem. More precisely, under the assumptions of invexity and Fréchet differentiability of the involved curvilinear integral functionals, and by using the notion of invex set with respect to some given functions, we derive the characterization results.</p></abstract>

An existence result for a generalized vector equilibrium problem is proved in a general topological vector space. As applications, existence results are derived for vector equilibrium problems and vector variational inequalities with set-valued mappings which have order-lower (upper) semicontinuous and pseudomonotone properties. These results extend and unify a number of existence results for Vector Variational Inequalities in the previous literature.

We consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities. Developing a simple Lagrangian duality scheme which is combined with the recent logarithmic-quadratic proximal (LQP) theory introduced by the authors, we derive three algorithms for solving the variational inequality (VI) problem. This provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms. We prove primal, dual, and primal-dual convergence under very mild assumptions, eliminating all the usual assumptions used until now in the literature for related algorithms. Applications to complementarity problems are also discussed.

The paper aims to show that a Vector Variational Inequality can be an useful tool for studying a Vector Optimization Problem.Key WordsVector variational inequalityvector optimization problemstrong monotonicitystrong convexitysolution setconnectednesscompactnessperturbationsensitivity

Weak sharp minima for set-valued vector variational inequalities with an application