A Simplex Method for Infinite Linear Programs

In this paper, we present a simplex method for linear programs in standard form, or more precisely, linear optimization problems that have countably many non-negative variables and countably many equality constraints. Important special cases of these problems include infinite horizon deterministic dynamic programming problems and network flow problems with countably infinite nodes and arcs. After embedding the primal linear program and its dual in appropriate topological dual spaces, we develop an algebraic characterization of extreme points in terms of the basic, i.e., strictly positive variables. Although such a characterization is hard to accomplish in the most general case, we show that it can be successfully developed when the infimum of the values of basic variables of an extreme point is strictly positive. An important case where this infimum condition is met is when the components of an extreme point are integers. We illustrate how this algebraic characterization can be used to develop important extensions of some well-known results in the finite variable, finite constraint case. We characterize degeneracy and discuss challenges involved in resolving it. We show that our simplex method maintains primal feasibility and complementary slackness in every iteration and achieves dual feasibility in the limit.