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 Haar's 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.