Two-Period Stochastic Programs with Simple Recourse

Stochastic programs are said to have simple recourse if the state vector in each period is uniquely determined once all previous decision and random vectors are known. This paper considers two-period problems of this nature. A number of important business and economic problems such as those concerned with inventory management, portfolio revision, cash balance management, and pension fund management can be formulated effectively as problems in this class. We present conditions that ensure that there is an equivalent deterministic convex program that has a directionally differentiable objective function. Detailed expressions enabling one to calculate the directional derivative are derived. Thus it is possible to use Hogan's modification of the Frank-Wolfe algorithm, which applies for two-stage convex programs, in the solution of these two-stage stochastic convex programs. Easily calculated bounds on the optimal objective value and the problem's Kuhn-Tucker conditions are presented. We consider the important special cases when there are no lagged state variables, discrete probability distributions, and direct reduction possibilities. We show that these cases, which frequently arise in practice, yield equivalent static problems. The problems of cash balance and pension fund management are treated in detail.