Duality results for a class of stochastic optimization problems with applications to an inventory control problem

Abstract

We consider a class of discrete time stochastic optimization problems with a finite horizon. The problem is defined on a filtered probability space where the control variables are predictable stochastic processes in L?-space of adapted processes. The dual optimization problem is derived by using the theory of conjugate functions and convex integral functionals. Despite the fact that the dual space of L? cannot be identified with L1, the main result that the dual variables are L1-martingales is proved. Then, the sufficient conditions for the existence of a dual solution and for the optimal values of the primal and dual problems to be equal are given by using a version of Fenchel Duality Theorem. We demonstrate how the previous results apply to an inventory control problem. The inventory control problem analyzed is to choose the amount of an item to be ordered at the beginning of each period so as to minimize the expected sum of ordering, holding and shortage costs, where all costs are assumed to be convex and increasing. We give necessary and sufficient conditions of optimality and relate the optimal primal and dual solutions. Finally, we show that using the optimal dual process as a price system leads to an efficient dual optimization algorithm by enabling us to replace the original problem with an easier one which is an important implication of this duality theory.