Catálogo de publicaciones - libros

Inspired by some results from nonsmooth critical point theory, we propose in this paper to study equilibrium problems by means of a general Palais-Smale condition adapted to bifunctions. We introduce the notion of critical points for equilibrium problems and we give some existence results for (EP) with lack of compacity.

Minty variational inequalities are studied as a tool for vector optimization. Instead of focusing on vector inequalities, we propose an approach through scalarization which allows to construct a proper variational inequality type problem to study any concept of efficiency in vector optimization.

This general scheme gives an easy and consistent extension of scalar results, providing also a notion of increasing along rays vector function. This class of generalized convex functions seems to be intimately related to the existence of solutions to a Minty variational inequality in the scalar case, we now extend this fact to vector case.

Finally, to prove a reversal of the main theorem, generalized quasiconvexity is considered and the notion of *-quasiconvexity plays a crucial role to extend scalar evidences. This class of functions, indeed, guarantees a Minty-type variational inequality is a necessary and sufficient optimality condition for several kind of efficient solution.

We consider the constrained vector optimization problem min(), () ∈ − , where : ℝ → ℝ and : ℝ → ℝ are given functions and ∈ ℝ and ∈ ℝ are closed convex cones. Two type of solutions are important for our considerations, namely -minimizers (isolated minimizers) of order and minimizers (properly efficient points) of order (see e.g. []). Every -minimizer of order ≥ 1 is a -minimizer of order . For = 1, conditions under which the reversal of this statement holds have been given in []. In this paper we investigate the possible reversal of the implication -minimizer ⇒ -minimizer in the case = 2. To carry on this study, we develop second-order optimality conditions for -minimizers, expressed by means of Dini derivatives. Together with the optimality conditions obtained in [] and [] in the case of -minimizers, they play a crucial role in the investigation. Further, to get a satisfactory answer to the posed reversal problem, we deal with sense I and sense II solution concepts, as defined in [] and [].

In terms of -th order Dini directional derivative with positive integer we define -pseudoconvex functions being a generalization of the usual pseudoconvex functions. Again with the -th order Dini derivative we define -stationary points, and prove that a point is a global minimizer of a -pseudoconvex function if and only if is a -stationary point of . Our main result is the following. A radially continuous function defined on a radially open convex set in a real linear space is -pseudoconvex if and only if is quasiconvex function and any -stationary point is a global minimizer. This statement generalizes the results of Crouzeix, Ferland, Math. Program. 23 (1982), 193–205, and Komlósi, Math. Program. 26 (1983), 232–237. We study also other aspects of the -pseudoconvex functions, for instance their relations to variational inequalities.

We consider a multiobjective optimization problem in ℝ with a feasible set defined by inequality and equality constraints and a set constraint. All the involved functions are, at least, directionally differentiable. We provide sufficient optimality conditions for global and local Pareto minimum under several kinds of generalized convexity. Also Wolfe-type and Mond-Weir-type dual problems are considered, and weak and strong duality theorems are proved.

In this work, approximate solutions of vector optimization problems in the sense of Tanaka [] are characterized via scalarization. Necessary and sufficient conditions are obtained using a new order representing property and a new monotonicity concept, respectively. A family of gauge functions defined by generalized Chebyshev norms and verifying both properties is introduced in order to characterize approximate solutions of vector optimization problems via approximate solutions of several scalarizations.

In this paper arcwise connected convex multifunctions are introduced and studied. Optimality conditions involving this type of data are analyzed.

Several kinds of continuous (generalized) monotone maps are characterized by partial gradient maps of skew-symmetric real-valued bifunctions displaying corresponding (generalized) concavity-convexity properties. As an economic application, it is shown that two basic approaches explaining consumer choice are behaviorally equivalent.

Necessary and sufficient optimality conditions for nonsmooth multiobjective optimization problems and some characterizations of convex vector functions are proved by means of mollified derivatives.

In this paper arcwise connected convex multifunctions are introduced and studied. Optimality conditions involving this type of data are analyzed.