Infinite Horizon Programs - Grinold, Richard C. (Management Science, 1971)
Several difficulties arise when multistage optimization problems with linear constraints are extended to an infinite horizon. For linear objectives the weak duality theorem fails, duality gaps exist, complementary slackness is not a sufficient condition for optimality, and the total reward may diverge. This paper presents verifiable conditions on the problem that guarantee the existence of optimal solutions and equilibrium prices for linear and nonlinear objectives. In addition, it is ascertained that the Kuhn-Tucker conditions are sufficient for optimality if additional restrictions are imposed. If the constraints are Leontief and the objective is convex (in a maximization problem), then the existence of an extreme point optimal solution can be verified. The decision taken in each period is an extreme point of the set of solutions that are feasible in that period. The Leontief case also yields a horizon theorem. All sufficiently long finite horizon problems produce a first decision that is optimal for the infinite horizon problem.