Two-Stage Stochastic Mixed-Integer Programs: Algorithms and Insights

Stochastic (mixed-) integer programs pose a great algorithmic and computational challenge in that they combine two generally difficult classes of problems: stochastic programs and discrete optimization problems. Exploring its dual angular structure, various decomposition methods have been widely studied, including Benders' decomposition, Lagrangian relaxation, and test-set decomposition. These decomposition methods are often combined with search procedures such as branch-and-bound or branch-and-cut. Within the confines of these broad frameworks, fine-tuned algorithms have been proposed to overcome obstacles such as nonconvexity of the second-stage value functions under integer recourse, and to take advantage of the many similar structured scenario subproblems using variable transformations. In this chapter, we survey some recent algorithms developed to solve two-stage stochastic (mixed-) integer programs, as well as provide insights into and results concerning their interconnections, particularly for alternative convexification techniques.