Primal-dual partitions in linear semi-infinite programming with bounded coefficients

Abstract

We consider two partitions over the space of linear semi-infinite programming parameters with a fixed index set and bounded coefficients (the constraint functions are bounded). The first one is the primal-dual partition inspired by consistency and boundedness of the optimal value of the problem. The second one is a refinement of the primal-dual partition that arises by considering also the boundedness of the optimal set. These two partitions have been studied in the continuous case, i.e., when the set of indices is an infinite compact topological space and the constraint functions are continuous. In this paper, we extend these results to the case in which the constraint functions are bounded, but not necessarily continuous. We study the same primal-dual partitions and characterize the interior of the corresponding cells. Through examples, we show that the conditions characterizing the cells of both partitions in the continuous case are neither necessary nor sufficient when the constraint functions are just bounded. In addition, a sufficient condition for the boundedness of the optimal set of the dual problem is established.