Existence and duality theory for separated continuous linear programs

Abstract

This paper surveys the recent developments in the theoretical study of separated continuous linear programs (SCLP). This problem serves as a useful model for various dynamic network problems where storage is permitted at the nodes. We demonstrate this by modelling some hypothetical problems of water distribution, transportation and telecommunications. The theoretical developments we present for SCLP fall into two main topics. The first of these is the existence of optimal solutions of various forms. These results culminate in one guaranteeing the existence of a piecewise analytic optimal solution, that is, having a finite number of breakpoints. The second topic we discuss is duality. Under this heading we develop a theory that closely resembles that for finite-dimensional linear programming. For instance, we define complementary slackness and give conditions under which there exist complementary slack primal and dual optimal solutions. Throughout the paper we observe that the main theorems are sufficiently general to include any reasonable practical problems