Maxim I. Todorov

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.

Stability and Well-Posedness in Linear Semi-Infinite Programming

This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.

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.

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.

Constraint qualifications in linear vector semi-infinite optimization

Linear vector semi-infinite optimization deals with the simultaneous minimization of finitely many linear scalar functions subject to infinitely many linear constraints. This paper provides characterizations of the weakly efficient, efficient, properly efficient and strongly efficient points in terms of cones involving the data and Karush–Kuhn–Tucker conditions. The latter characterizations rely on different local and global constraint qualifications. The global constraint qualifications are illustrated on a collection of selected applications.

Primal-dual stability in continuous linear optimization

Any linear (ordinary or semi-infinite) optimization problem, and also its dual problem, can be classified as either inconsistent or bounded or unbounded, giving rise to nine duality states, three of them being precluded by the weak duality theorem. The remaining six duality states are possible in linear semi-infinite programming whereas two of them are precluded in linear programming as a consequence of the existence theorem and the non-homogeneous Farkas Lemma. This paper characterizes the linear programs and the continuous linear semi-infinite programs whose duality state is preserved by sufficiently small perturbations of all the data. Moreover, it shows that almost all linear programs satisfy this stability property.

Extended Active Constraints in Linear Optimization with Applications

In this paper we introduce different relaxations of the concept of active constraint at a given point with respect to a certain linear inequality system with an arbitrary number of constraints. We show that these concepts provide useful local information in linear optimization, for instance conditions for a given feasible solution to be a unique, extreme point, optimal solution or a strongly unique optimal solution.

Generic primal-dual solvability in continuous linear semi-infinite programming

In this article, we consider the space of all the linear semi-infinite programming (LSIP) problems with a given infinite compact Hausdorff index set, a given number of variables and continuous coefficients, endowed with the topology of the uniform convergence. These problems are classified as inconsistent, solvable with bounded optimal set, bounded (i.e. finite valued), but either unsolvable or having an unbounded optimal set, and unbounded (i.e. with infinite optimal value), giving rise to the so-called refined primal partition of the space of problems. The mentioned LSIP problems can be also classified with a similar criterion applied to the corresponding Haars dual problems, which provides the refined dual partition of the space of problems. We characterize the interior of the elements of the refined primal and dual partitions as well as the interior of the intersections of the elements of both partitions (the so-called refined primal-dual partition). These characterizations allow to prove that most (primal or dual) bounded problems have simultaneously primal and dual non-empty bounded optimal set. Consequently, most bounded continuous LSIP problems are primal and dual solvable. †In celebration of Prof. Dr Hubertus Th. Jongens 60th Birthday.

Primal, dual and primal-dual partitions in continuous linear optimization

We associate with each natural number n and each compact Hausdorff topological space T the space of linear optimization problems with n primal variables and index set T (for the constraints) equipped with the topology of the uniform convergence. We consider three different partitions of this metric space. The primal and the dual partitions are the result of classifying a given optimization problem and its dual as either inconsistent or bounded or unbounded, whereas the primal-dual partition is formed by the nonempty intersections of the elements of both partitions. The elements of the three partitions are neither open nor closed and their topological interiors are formed by those problems for which sufficiently small perturbations maintain the membership of the problem, i.e. the problems that are stable for the corresponding property. We prove that the stable problems are the same for the three partitions, concluding that most problems are stable in the three senses. This is done by completing the topological analysis of the primal-dual partition carried out in a previous paper of the authors.

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.