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

Abstract

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.