G. Ruiz-Garzón

Relationships between vector variational-like inequality and optimization problems

Abstract In this paper we will establish the relationships between vector variational-like inequality and optimization problems. We will be able to identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality problem, under conditions of pseudo invexity. These conditions are more general those existing in the literature.

Generalized invex monotonicity

Abstract In this paper the generalized invex monotone functions are defined as an extension of monotone functions. A series of sufficient and necessary conditions are also given that relate the generalized invexity of the function θ with the generalized invex monotonicity of its gradient function ∇θ. This new class of functions will be important in order to characterize the solutions of the Variational-like Inequality Problem and Mathematical Programming Problem.

KT-invex control problem

Abstract In this paper, we introduce a new condition on functions of a control problem, for which we define a KT-invex control problem. We prove that a KT-invex control problem is characterized in order that a Kuhn–Tucker point is an optimal solution. We generalize optimality results of known mathematical programming problems. We illustrate these results with examples.

On invex fuzzy mappings and fuzzy variational-like inequalities

In this paper, we first show the need for introducing invex fuzzy mappings. After that, we show that the concept of invex fuzzy mapping previously given by Wu and Xu are very restrictive and the examples presented there are not correct. Then, we present more general concepts of invex and incave fuzzy mappings involving strongly generalized differentiable fuzzy mapping. Finally, we show that the results obtained by Wu and Xu on the relationship between fuzzy variational-like inequalities and fuzzy optimization problems are still valid using these new concepts.

Existence of weakly efficient solutions in nonsmooth vector optimization

In this paper we study the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems. We consider problems whose objective functions are defined between infinite and finite-dimensional Banach spaces. Our results are stated under hypotheses of generalized convexity and make use of variational-like inequalities.

Weak efficiency in multiobjective variational problems under generalized convexity

In this paper, we provide new pseudoinvexity conditions on the involved functionals of a multiobjective variational problem, such that all vector Kuhn-Tucker or Fritz John points are weakly efficient solutions if and only if these conditions are fulfilled. We relate weakly efficient solutions to optimal solutions of weighting problems. We improve recent papers, and we generalize pseudoinvexity conditions used in multiobjective mathematical programming, so as some of their characterization results. The new conditions and results are illustrated with an example.

A characterization of pseudoinvexity in multiobjective programming

In this paper, we introduce new classes of vector functions which generalize the class of scalar invex functions. We prove that these new classes of vector functions are characterized in such a way that every vector critical point is an efficient solution of a Multiobjective Programming Problem. We establish relationships between these new classes of functions and others used in the study of efficient and weakly efficient solutions.

Some relations between variational-like inequality problems and vectorial optimization problems in Banach spaces

In this work, we will establish some relations between variational-like inequality problems and vectorial optimization problems in Banach spaces under invexity hypotheses. This paper extends the earlier work of Ruiz-Garzon et al. [G. Ruiz-Garzon, R. Osuna-Gomez, A. Rufian-Lizana, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004) 113-119].

FJ-Invex control problem

Abstract This paper introduces a new condition on the functionals of a control problem and extends a recent characterization result of KT-invexity. We prove that the new condition, the FJ-invexity, is both necessary and sufficient in order to characterize the optimal solution set using Fritz John points.

Duality and a Characterization of Pseudoinvexity for Pareto and Weak Pareto Solutions in Nondifferentiable Multiobjective Programming

In this paper, we unify recent optimality results under directional derivatives by the introduction of new pseudoinvex classes of functions, in relation to the study of Pareto and weak Pareto solutions for nondifferentiable multiobjective programming problems. We prove that in order for feasible solutions satisfying Fritz John conditions to be Pareto or weak Pareto solutions, it is necessary and sufficient that the nondifferentiable multiobjective problem functions belong to these classes of functions, which is illustrated by an example. We also study the dual problem and establish weak, strong, and converse duality results.