Variational Inequalities and Vector Optimization 2014

Abstract. In this paper, we consider and study a new class of general-ized vector quasi-variational type inequalities and obtain some existencetheorems for both under compact and noncompact assumptions in topo-logical vector spaces without using monotonicity. For the noncompactcase, we use the concept of escaping sequences. 1. IntroductionThe vector variational inequality problem was initiated by Giannessi [11] in nite dimensional Euclidean spaces with applications. Later on many authors[6, 7, 12, 15, 24] generalized vector variational inequalities in abstract spacesin several ways. The vector variational-like inequality is one of the generalizedforms of vector variational inequalities [11]. In 1989, Parida et al [19] stud-ied the existence of solutions for variational-like inequalities in R n space andhave shown a relationship between variational like inequalities problems andconvex programming as well as with complementarity problems. The vectorvariational like inequality and generalized vector variational like inequality arepowerful tools to study non-convex vector optimization problems and convexand nondi erentiable vector optimization problems respectively, see [10, 18, 21].In 1973, Bensoussan and Lions [3] introduced the quasi-variational inequalityproblem (QVIP). Since then, many generalizations of the QVIP have beenappeared in the literature (see, for example, [1, 2]). In the last decade, becauseof applications in the optimization problems, mathematical programming andequilibrium problems, the QVIP has been intensively studied by many authors[5, 8, 16, 20, 22, 23, 25].In this paper, we consider the generalized vector quasivariational type in-equality problem (GVQVTIP) and obtain some existence theorems for solutions

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

Invited Papers.- Combined Relaxation Methods for Generalized Monotone Variational Inequalities.- Abstract Convexity and the Monge-Kantorovich Duality.- Optimality Conditions and Duality for Multiobjective Programming Involving (C, ?, ?, d) type-I Functions.- Contributed Papers.- Partitionable Variational Inequalities with Multi-valued Mappings.- Almost Convex Functions: Conjugacy and Duality.- Pseudomonotonicity of a Linear Map on the Interior of the Positive Orthant.- An Approach to Discrete Convexity and Its Use in an Optimal Fleet Mix Problem.- A Unifying Approach to Solve a Class of Parametrically-Convexifiable Problems.- Mathematical Programming with (?, ?)-invexity.- Some Classes of Pseudoconvex Fractional Functions via the Charnes-Cooper Transformation.- Equilibrium Problems Via the Palais-Smale Condition.- Points of Efficiency in Vector Optimization with Increasing-along-rays Property and Minty Variational Inequalities.- Higher Order Properly Efficient Points in Vector Optimization.- Higher-order Pseudoconvex Functions.- Sufficient Optimality Conditions and Duality in Nonsmooth Multiobjective Optimization Problems under Generalized Convexity.- Optimality Conditions for Tanaka's Approximate Solutions in Vector Optimization.- On the Work of W. Oettli in Generalized Convexity and Nonconvex Optimization - a Review and Some Perspectives.- Local and Global Consumer Preferences.- Optimality Conditions for Convex Vector Functions by Mollified Derivatives.- On Arcwise Connected Convex Multifunctions.- A Sequential Method for a Class of Bicriteria Problems.- Decomposition of the Measure in the Integral Representation of Piecewise Convex Curves.- Rambling Through Local Versions of Generalized Convex Functions and Generalized Monotone Operators.- Monotonicity and Dualities.- On Variational-like Inequalities with Generalized Monotone Mappings.- Almost Pure Nash Equilibria in Convex Noncooperative Games.- A Spectral Approach to Solve Box-constrained Multi-objective Optimization Problems.

<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>

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.

Robust counterparts and robust efficient solutions in vector optimization under uncertainty

Equilibrium is a concept central to the analysis of economic phenomena. Methodologies that have been applied to the formulation, qualitative analysis, and computation of economic equilibria have included systems of equations, optimization theory, complementarity theory, as well as fixed point theory. In this chapter the foundations for the theory of variational inequalities are established and the relationship of this methodology to other existing equilibrium analysis tools identified. Variational inequality theory will be utilized throughout the book as the fundamental methodology in synthesizing network economic equilibrium models operating under a spectrum of behavioral mechanisms and ranging from spatial price equilibrium problems and imperfectly competitive oligopolistic market equilibrium problems to general financial equilibrium problems.KeywordsVariational InequalityEquilibrium ProblemComplementarity ProblemVariational Inequality ProblemMonotonicity ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We take into condideration necessary optimality conditions of minimum principle-type, that is for optimization problems having, besides the usual inequality and/or equality constraints, a set constraint. The first part pf the paper is concerned with scalar optimization problems; the second part of the paper deals with vector optimization problems.

Surrogate duality for robust optimization

Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings

We use asymptotic analysis to develop finer estimates for the efficient, weak efficient and proper efficient solution sets (and for their asymptotic cones) to convex/quasiconvex vector optimization problems. We also provide a new representation for the efficient solution set without any convexity assumption, and the estimates involve the minima of the linear scalarization of the original vector problem. Some new necessary conditions for a point to be efficient or weak efficient solution for general convex vector optimization problems, as well as for the nonconvex quadratic multiobjective optimization problem, are established.

An iterative procedure for vector optimization

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

In this paper, the exact minimax penalty function method is applied to solve constrained multiobjective optimization problems involving locally Lipschitz functions. The criteria for a saddle point for the original vector optimization problem are studied with the help of the penalized unconstrained vector optimization problem. Furthermore, we determine the conditions for which the (weak) efficient solutions of the vector optimization problem are equivalent to those of the associated, penalized unconstrained vector optimization problem. Some examples of nonsmooth multiobjective problems solved by using the exact minimax penalty method are presented to illustrate the results established in the paper.