Optimized scenario reduction : solving large-scale stochastic programs with quality guarantees

Abstract

Stochastic programming involves large-scale optimization with exponentially many scenarios. This paper proposes an optimization-based scenario reduction approach to generate high-quality solutions and tight lower bounds by only solving small-scale instances, with a limited number of scenarios. First, we formulate a scenario subset selection model that optimizes the recourse approximation over a pool of solutions. We provide a theoretical justification of our formulation, and a tailored heuristic to solve it. Second, we propose a scenario assortment optimization approach to compute a lower bound—hence, an optimality gap—by relaxing nonanticipativity constraints across scenario “bundles.” To solve it, we design a new column-evaluation-and-generation algorithm, which provides a generalizable method for optimization problems featuring many decision variables and hard-to-estimate objective parameters. We test our approach on stochastic programs with continuous and mixed-integer recourse. Results show that (i) our scenario reduction method dominates scenario reduction benchmarks, (ii) our scenario assortment optimization, combined with column-evaluation-and-generation, yields tight lower bounds, and (iii) our overall approach results in stronger solutions, tighter lower bounds, and faster computational times than state-of-the-art stochastic programming algorithms.