The Set of Target Vectors in a Semi-Infinite Linear Program with a Duality Gap

Abstract

We propose a geometric method for the analysis of duality relations in a pair of semi-infinite linear programs (SILPs). The method is based on the use of the conic hull of the coefficients in the constraint system. A relation between the presence of a duality gap and the nonclosedness of the boundary of the conic hull of points in a multidimensional space is established. The geometric approach is used to construct an opposite pair of dual problems and to explore the duality relations for this pair. We construct a nontrivial example of a SILP in which the duality gap occurs for noncollinear target vectors.