Miguel A. Goberna

Stability Theory for Linear Inequality Systems

This paper develops a stability theory for (possibly infinite) linear inequality systems defined on a finite-dimensional space, analyzing certain continuity properties of the solution set mapping. It also provides conditions under which sufficiently small perturbations of the data in a consistent (inconsistent) system produce systems belonging to the same class.

Linear semi-infinite programming theory: An updated survey

Abstract This paper presents an state-of-the-art survey on linear semi-infinite programming theory and its extensions (in particular, convex semi-infinite programming). This review updates a previous survey [Semi-Infinite Programming, Non-convex Optim. Appl. 25, 1998] of the same authors on the same topic which was published in 1998.

Stability Theory for Linear Inequality Systems II: Upper Semicontinuity of the Solution Set Mapping

This paper deals with the upper semicontinuity of the solution set mapping for linear inequality systems, complementing a previous work on lower semicontinuity and related stability concepts. The main novelty of our approach is that we are not assuming any standard hypothesis about the set indexing the inequalities in the system. This set, possibly infinite, has no topological structure and, therefore, the functional dependence between the linear inequalities and their associated indices has no qualification at all. The space of consistent systems, over a fixed index set, is endowed with the uniform topology derived from the pseudometric of Chebyshev, which turns out to be a natural way to measure the size of the perturbations. In this context, we provide some necessary and some sufficient conditions for the upper semicontinuity of the feasible set map at a given system whose solution set is not necessarily bounded.

Robust solutions to multi-objective linear programs with uncertain data

In this paper we examine multi-objective linear programming problems in the face of data uncertainty both in the objective function and the constraints. First, we derive a formula for the radius of robust feasibility guaranteeing constraint feasibility for all possible scenarios within a specified uncertainty set under affine data parametrization. We then present numerically tractable optimality conditions for minmax robust weakly efficient solutions, i.e., the weakly efficient solutions of the robust counterpart. We also consider highly robust weakly efficient solutions, i.e., robust feasible solutions which are weakly efficient for any possible instance of the objective matrix within a specified uncertainty set, providing lower bounds for the radius of highly robust efficiency guaranteeing the existence of this type of solutions under affine and rank-1 objective data uncertainty. Finally, we provide numerically tractable optimality conditions for highly robust weakly efficient solutions.

Farkas-Minkowski systems in semi-infinite programming

The Farkas-Minkowski systems are characterized through a convex cone associated to the system, and some sufficient conditions are given that guarantee the mentioned property. The role of such systems in semi-infinite programming is studied in the linear case by means of the duality, and, in the nonlinear case, in connection with optimality conditions. In the last case the property appears as a constraint qualification.

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Sensitivity analysis in linear semi-infinite programming: Perturbing cost and right-hand-side coefficients

This paper analyzes the effect on the optimal value of a given linear semi-infinite programming problem of the kind of perturbations which more frequently arise in practical applications: those which affect the objective function and the right-hand-side coefficients of the constraints. In particular, we give formulae which express the exact value of a perturbed problem as a linear function of the perturbation.

On the Stability of the Boundary of the Feasible Set in Linear Optimization

This paper analizes the relationship between the stability properties of the closed convex sets in finite dimensions and the stability properties of their corresponding boundaries. We consider a given closed convex set represented by a certain linear inequality system σ whose coefficients can be arbitrarily perturbed, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. It is shown that the feasible set mapping is Berge lower semicontinuous at σ if and only if the boundary mapping satisfies the same property. Moreover, if the boundary mapping is semicontinuous in any sense (lower or upper; Berge or Hausdorff) at σ, then it is also closed at σ. All the mentioned stability properties are equivalent when the feasible set is a convex body.

Topological stability of linear semi-infinite inequality systems

In this note, we analyze the relationship between the lower semicontinuity of the feasible set mapping for linear semi-infinite inequality systems and the so-called topological stability, which is held when the solution sets of all the systems obtained by sufficiently small perturbations of the data are homeomorphic to each other. This topological stability and its relation with the Mangasarian-Fromovitz constraints qualification have been studied deeply by Jongen et al. in Ref. 1. The main difference of our approach is that we are not assuming any kind of structure for the index set and, consequently, any particular property for the functional dependence between the inequalities and the associated indices. In addition, we deal with systems whose solution sets are not necessarily bounded.

Dual Characterizations of Set Containments with Strict Convex Inequalities

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments.