On optimality conditions for vector variational inequalities (original) (raw)

Vector optimization and variational-like inequalities

2009

In this paper, some properties of pseudoinvex functions are obtained. We study the equivalence between different solutions of the vector variational-like inequality problem. Some relations between vector variational-like inequalities and vector optimization problems for non-differentiable functions under generalized monotonicity are established.

Vector nonsmooth variational-like inequalities and optimization problems

Nonlinear Analysis: Theory, Methods & Applications, 2009

In this paper, we first extend the concepts of Clarke's generalized derivation and subdifferential for a function whose domain is a subset of a metrizable topological vector space. We also introduce and consider a new class of variational-like inequalities, which are called the vector nonsmooth variational-like inequalities. We also establish the relationship between the nonsmooth variational-like inequalities and vector optimization problems under some suitable conditions. In particular, our results extend and generalize the results of Mishra and Wang [

Generalized vector variational-like inequalities and vector optimization

2012

In this paper, we consider different kinds of generalized vector variational-like inequality problems and a vector optimization problem. We establish some relationships between the solutions of generalized Minty vector variational-like inequality problem and an efficient solution of a vector optimization problem. We define a perturbed generalized Stampacchia vector variational-like inequality problem and discuss its relation with generalized weak Minty vector variational-like inequality problem. We establish some existence results for solutions of our generalized vector variational-like inequality problems.

On vector variational-like inequality problems

Journal of Mathematical Analysis and Applications, 2005

In this paper, we establish some relationships between vector variational-like inequality and vector optimization problems under the assumptions of α-invex functions. We identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality problems, under pseudo-α-invexity assumptions. These conditions are more general than those of existing ones in the literature. In particular, this work extends the earlier work of Ruiz-Garzon et al. [G. Ruiz-Garzon, R. Osuna-Gomez, A. Rufian-Lizan, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004) 113-119] to a wider class of functions, namely the pseudo-α-invex functions studied in a recent work of Noor [M.A. Noor, On generalized preinvex functions and monotonicities,

The regularity of some vector-valued variational inequalities with gradient constraints

Communications on Pure and Applied Analysis

We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar and vector-valued case.

Nonsmooth Multiobjective Problems and Generalized Vector Variational Inequalities Using Quasi-Efficiency

Journal of Optimization Theory and Applications, 2017

In this paper, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Here, a generalized Stampacchia vector variational inequality is formulated as a tool to characterize quasi-or weak quasi-efficient points. By using two new classes of generalized convexity functions, under suitable constraint qualifications, the equivalence between Kuhn-Tucker vector critical points, solutions to the multiobjective problem and solutions to the generalized Stampacchia vector variational inequality in both weak and strong forms will be proved. Keywords Nonsmooth multiobjective problems • Generalized Stampacchia vector variational inequalities • Weak quasi-efficiency • Kuhn-Tucker vector critical points • Constraint qualifications.

Necessary optimality conditions for optimization problems with variational inequality constraints

Mathematics of Operations Research, 1997

A very general optimization problem with a variational inequality constraint, inequality constraints and an abstract constraint is studied. Fritz John type and Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are derived. Several constraint quali cations for the Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are introduced and their relationships are studied. Applications to bilevel programming problems are also given.