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

Abstract

We associate with each natural number n and each compact Hausdor¤ 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 di¤erent partitions of this pseudometric 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 non-empty 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 su¢ -ciently 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.

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

“…Finally, Π P U B = Π 2 is non-open by Lemma 2.4(ii) and the same applies to Π P IC by [13,Proposition 2]. The proof of the next result is similar to the last one and will be omitted.…”

“…This class of primal-dual ill-posed parameters is, in our setting, bd Π 1 . The following lemma summarizes results on the primal-dual partition which appeared in [22] (where int Π 1 was characterized), [3], [12, Section 4] (taking into account that N can be replaced with K in all the characterizations) and [13]. …”

“…We are concerned with those parameters which are stable for the corresponding property in the sense that sufficiently small perturbations of the parameter preserve its membership. The interior of Π P C , Π P IC , Π D C , and Π D IC where characterized in [9] and [10], whereas [3], [12] and [13] have characterized the interior of the elements of the primal partition (of Π), Π P IC , Π P B , Π P U B , the dual partition,…”