Virginia N. Vera de Serio

Locally Farkas-Minkowski linear inequality systems

This paper introduces thelocally Farkas-Minkowski (LFM) linear inequality systems in a finite dimensional Euclidean space. These systems are those ones that satisfy that any consequence of the system that is active at some solution point is also a consequence of some finite subsystem. This class includes the Farkas-Minkowski systems and verifies most of the properties that these systems possess. Moreover, it contains the locally polyhedral systems, which are the natural external representation of quasi-polyhedral sets. TheLFM systems appear to be the natural external representation of closed convex sets. A characterization based on their properties under the union of systems is provided. In linear semi-infinite programming, theLFM property is the more general constraint qualification such that the Karush-Kuhn-Tucker condition characterizes the optimal points. Furthermore, the pair of Haar dual problems has no duality gap.

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.

Stability of the Feasible Set Mapping in Convex Semi-Infinite Programming

In this paper we approach the stability analysis of the feasible set mapping in convex semi-infinite programming for an arbitrary index set. More precisely, we establish its closedness and study the semicontinuity, in the sense of Berge, of this multivalued mapping- A certain metric is proposed in order to measure the distance between nominal and perturbed problems. Since we do not require any structure to the index set, our results cover the ordinary convex programming problem.

Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems

We consider convex semiinfinite programming (SIP) problems with an arbitrary fixed index set T. The article analyzes the relationship between the upper and lower semicontinuity (lsc) of the optimal value function and the optimal set mapping, and the so-called Hadamard well-posedness property (allowing for more than one optimal solution). We consider the family of all functions involved in some fixed optimization problem as one element of a space of data equipped with some topology, and arbitrary perturbations are premitted as long as the perturbed problem continues to be convex semiinfinite. Since no structure is required for T, our results apply to the ordinary convex programming case. We also provide conditions, not involving any second order optimality one, guaranteeing that the distance between optimal solutions of the discretized subproblems and the optimal set of the original problem decreases by a rate which is linear with respect to the discretization mesh-size.

Stability of the primal-dual partition in linear semi-infinite programming

We analyse the primal-dual states in linear semi-infinite programming (LSIP), where we consider the primal problem and the so called Haars dual problem. Any linear programming problem and its dual can be classified as bounded, unbounded or inconsistent, giving rise to nine possible primal-dual states, which are reduced to six by the weak duality property. Recently, Goberna and Todorov have studied this partition and its stability in continuous LSIP in a series of papers [M.A. Goberna and M.I. Todorov, Primal, dual and primal-dual partitions in continuous linear semi-infinite programming, Optimization 56 (2007), pp. 617–628; M.A. Goberna and M.I. Todorov, Generic primal-dual solvability in continuous linear semi-infinite programming, Optimization 57 (2008), pp. 239–248]. In this article we consider the general case, with no continuity assumptions, discussing the maintenance of the primal-dual state of the problem by allowing small perturbations of the data. We characterize the stability of all of the six possible primal-dual states through necessary and sufficient conditions which depend on the data, and can be easily checked, showing some differences with the continuous case. These conditions involve the strong Slater constraint qualification, and some distinguished convex sets associated to the data.

On the Stability of the Extreme Point Set in Linear Optimization

This paper analyzes the stability properties of the set of extreme points of a closed convex set described by means of a given linear inequality system

On stable uniqueness in linear semi-infinite optimization

\sigma

Quasipolyhedral sets in linear semiinfinite inequality systems

. We assume that all the coefficients of

On the stable containment of two sets

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Pricing, learning, and strategic behavior in a single-sale model

can be arbitrarily perturbed maintaining the (possibly infinite) index set as well as the (finite) dimension of the space of variables, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. The paper characterizes the nonemptiness of the extreme point set under sufficiently small perturbations and the Berge lower semicontinuity of the extreme point set mapping at