Duality and sensitivity analysis of multistage linear stochastic programs

Abstract

In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP). By writing Dynamic Programming equations for the dual, we can employ an SDDP type method, called Dual SDDP, which solves these Dynamic Programming equations. allows us to compute a sequence of nonincreasing deterministic upper bounds for the optimal value of the problem. Since the Relatively Complete Recourse (RCR) condition may fail to hold for the dual (even for simple problems), we design two variants of Dual SDDP, namely Dual SDDP with penalizations and Dual SDDP with feasibility cuts, that converge to the optimal value of the dual (and therefore primal when there is no duality gap) problem under mild assumptions. We also show that optimal dual solutions can be obtained using dual information from Primal SDDP (applied to the original primal MSLP) subproblems.As a byproduct of the developed methodology we study sensitivity of the optimal value of the problem as a function of the involved parameters. For the sensitivity analysis we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them, i.e., we need to solve the dual problem.Finally, as a proof of concept of the tools developed, we present the results of numerical experiments computing the sensitivity of the optimal value of an inventory problem as a function of parameters of the demand process and compare Primal and Dual SDDP on the inventory and a hydro-thermal planning problems.