Boundedness relations in linear semi-infinite programming

AbstractThis paper gives theorems on the boundedness of the feasible and the optimal solutions sets of a dual pair of linear semi-infinite programs (in Haar's duality). It also provides conditions for the boundedness of the primal slack variables and dual structural variables

On the stability of the optimal value and the optimal set in optimization problems

The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in [17], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property)

Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

Stability of the feasible set mapping of linear systems with an exact constraint set

This paper deals with the stability of the feasible set mapping of linear systems of an arbitrary number (possibly infinite) of equations and inequalities such that the variable x ranges on a certain fixed constraint set X⊂ℝn (X could represent the solution set of a given constraint system, e.g., the positive cone of ℝn in the case of sign constraints). More in detail, the paper provides necessary as well as sufficient conditions for the lower and upper semicontinuity (in Berge sense), and the closedness, of the set-valued mapping which associates, with any admissible perturbation of the given (nominal) system its feasible set. The parameter space is formed by all the systems having the same structure (i.e., the same number of variables, equations and inequalities) as the nominal one, and the perturbations are measured by means of the pseudometric of the uniform convergence.Partially supported by Fondecyt Grant 1020(7020)-646 and DI-U. de Chile Grant ENL 06/11, and by MEC and FEDER, Grant MTM2005-08572-C03-01

On stable uniqueness in linear semi-infinite optimization

This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-infinite optimization (LSIO) problem, in the sense of maintaining the strong uniqueness property under sufficiently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (affinely) stable problems, i.e., those LSIO problems whose feasible set (its affine hull, respectively) remains constant under sufficiently small perturbations.This work has been supported by MICINN of Spain, Grant MTM2008-06695-C03-01/03, by Generalitat Valenciana, by CONACyT of MX, Grant 55681, and by SECTyP-UNCuyo Res. 882/07-R

Uniform saturation in linear inequality systems

Linear systems, saturation, redundancy, linear optimization, 15A39, 90C05, 90C34,

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.This work has been supported by MICINN of Spain, Grants MTM2008-06695-C03-01/03, by Generalitat Valenciana, by Generalitat de Catalunya, by the Barcelona GSE Research Network, and by CONACyT of Mexico, Grant 55681

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.This work has been supported by MICINN of Spain, Grant MTM2008-06695-C03-01/03 and SECTyP-UNCuyo, Argentina

Revista de enseñanza universitaria

Resumen tomado de la revistaSe describe una experiencia, llevada a cabo en la Facultad de Ciencias de la Universidad de Alicante (Sección de Ciencias Químicas), que muestra hasta qué punto persiste la eficiencia académica en el tránsito de un nivel educativo a otro. Se analizan también las raíces ocultas de este fenómeno.AndalucíaES