Catálogo de publicaciones - libros

The paper is devoted to the combined relaxation approach to constructing solution methods for variational inequalities. We describe the basic idea of this approach and implementable methods both for single-valued and for multi-valued problems. All the combined relaxation methods are convergent under very mild assumptions. This is the case if there exists a solution to the dual formulation of the variational inequality problem. In general, these methods attain a linear rate of convergence. Several classes of applications are also described.

In the present survey, we reveal links between abstract convex analysis and two variants of the Monge-Kantorovich problem (MKP), with given marginals and with a given marginal difference. It includes: (1) the equivalence of the validity of duality theorems for MKP and appropriate abstract convexity of the corresponding cost functions; (2) a characterization of a (maximal) abstract cyclic monotone map : → ⊂ IR in terms connected with the constraint set of a particular dual MKP with a given marginal difference and in terms of -subdifferentials of -convex functions; (3) optimality criteria for MKP (and Monge problems) in terms of abstract cyclic monotonicity and non-emptiness of the constraint set (), where is a special cost function on × determined by the original cost function on × . The Monge-Kantorovich duality is applied then to several problems of mathematical economics relating to utility theory, demand analysis, generalized dynamics optimization models, and economics of corruption, as well as to a best approximation problem.

In this chapter, we present a unified formulation of generalized convex functions. Based on these concepts, sufficient optimality conditions for a nondifferentiable multiobjective programming problem are presented. We also introduce a general Mond-Weir type dual problem of the problem and establish weak duality theorem under generalized convexity assumptions. Strong duality result is derived using a constraint qualification for nondifferentiable multiobjective programming problems.

We consider multi-valued variational inequalities defined on a Cartesian product of finite-dimensional subspaces. We introduce extensions of order monotonicity concepts for set-valued mappings, which are adjusted to the case where the subspaces need not be real lines. These concepts enable us to establish new existence and uniqueness results for the corresponding partitionable multi-valued variational inequalities. Following a parametric coercivity approach, we obtain convergence of the Tikhonov regularization method without monotonicity conditions.

We prove that the formulae of the conjugates of the precomposition with a linear operator, of the sum of finitely many functions and of the sum between a function and the precomposition of another one with a linear operator hold even when the convexity assumptions are replaced by almost convexity or nearly convexity. We also show that the duality statements due to Fenchel hold when the functions involved are taken only almost convex, respectively nearly convex.

In this paper we will establish some necessary and/or sufficient conditions for both a nonsingular and a singular matrix (interpreted as a linear map) to be pseudomonotone. The given results are in terms of the sign of the determinants of the principal submatrices and of the cofactors of in the nonsingular case and in terms of the structure of in the singular case. A complete characterization of pseudomonotonicity in terms of the coefficients of a 3 × 3 matrix is given and a method for constructing a merely pseudomonotone matrix is suggested.

A notion of convexity for discrete functions is first introduced, with the aim to guarantee both the increasing monotonicity of marginal increments and the convexity of the sum of convex functions. Global optimality of local minima is then studied both for single variable functions and for multi variables ones. Finally, a concrete optimal fleet mix problem is studied, pointing out its discrete convexity properties.

The aim of this paper is to show how a wide class of generalized quadratic programs can be solved, in a unifying framework, by means of the so-called optimal level solutions method. In other words, the problems are solved by analyzing, explicitly or implicitly, the optimal solutions of particular quadratic strictly convex parametric subproblems. In particular, it is pointed out that some of these problems share the same set of optimal level solutions. A solution algorithm is proposed and fully described. The results achieved are then deepened in the particular case of box constrained problems.

We introduce new invexity-type properties for differentiable functions, generalizing ()-convexity. Optimality conditions for nonlinear programming problems are established under such assumptions, extending previously known results. Wolfe and Mond-Weir duals are also considered, and we obtain direct and converse duality theorems.

Using a very recent approach based on the Charnes-Cooper trasformation we characterize the pseudoconvexity of the sum between a quadratic fractional function and a linear one. Furthemore we prove that the ratio between a quadratic fractional function and the cube of an affine one is pseudoconvex if and only if the product between a quadratic fractional function and an affine one is pseudoconvex and we provide a sort of canonical form for this latter class of functions. Benefiting by the new results we are able to characterize the pseudoconvexity of the ratio between a quadratic fractional function and the cube of an affine one.