Nested Decomposition of Multistage Stochastic Integer Programs with Binary State Variables

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) that proceed by iteratively approximating these functions by cuts or linear inequalities, have been established as effective approaches. It is difficult to directly adapt these algorithms to MSIP due to the nonconvexity of integer programming value functions. In this paper, we propose a valid nested decomposition algorithm for MSIP when the state variables are restricted to be binary. We prove finite convergence of the algorithm as long as the cuts satisfy some sufficient conditions. We discuss the use of well known Benders’ and integer optimality cuts within this algorithm, and introduce new cuts derived from a Lagrangian relaxation corresponding to a reformulation of the problem where local copies of state variables are introduced. We propose a stochastic version of the nested decomposition algorithm and prove its finite convergence with probability one. In the case of stage-wise independent uncertainties this stochastic algorithm provides an extension of the SDDP approach for MSIP with binary state variables. Finally, 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 may lead to significant improvement on solving large-scale, multistage stochastic optimization problems in real-world applications.