Stochastic dual dynamic integer programming

Abstract

Multistage stochastic integer programming (MSIP) combines the difficulty of uncertainty, dynamics, and non-convexity, and constitutes a class of extremely challenging problems. A common formulation for these problems is a dynamic programming formulation involving nested cost-to-go functions. In the linear setting, the cost-to-go functions are convex polyhedral, and decomposition algorithms, such as nested Benders’ decomposition and its stochastic variant, stochastic dual dynamic programming (SDDP), which proceed by iteratively approximating these functions by cuts or linear inequalities, have been established as effective approaches. However, it is difficult to directly adapt these algorithms to MSIP due to the nonconvexity of integer programming value functions. In this paper we propose an extension to SDDP—called stochastic dual dynamic integer programming (SDDiP)—for solving MSIP problems with binary state variables. The crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts, termed Lagrangian cuts, derived from a Lagrangian relaxation of a specific reformulation of the subproblems in each stage, where local copies of state variables are introduced. We show that the Lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of SDDiP with probability one. We show that, under fairly reasonable assumptions, an MSIP problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size. Thus our proposed SDDiP approach is applicable to very general classes of MSIP problems. Extensive computational experiments on three classes of real-world problems, namely electric generation expansion, financial portfolio management, and network revenue management, show that the proposed methodology is very effective in solving large-scale multistage stochastic integer optimization problems.