Optimization-Driven Scenario Grouping

Abstract

Scenario decomposition algorithms for stochastic programs compute bounds by dualizing all nonanticipativity constraints and solving individual scenario problems independently. We develop an approach that improves on these bounds by reinforcing a carefully chosen subset of nonanticipativity constraints, effectively placing scenarios into groups. Specifically, we formulate an optimization problem for grouping scenarios that aims to improve the bound by optimizing a proxy metric based on information obtained from evaluating a subset of candidate feasible solutions. We show that the proposed grouping problem is NP-hard in general, identify a polynomially solvable case, and present two formulations for solving the problem: a matching formulation for a special case and a mixed-integer programming formulation for the general case. We use the proposed grouping scheme as a preprocessing step for a particular scenario decomposition algorithm and demonstrate its effectiveness in solving standard test instances of two-stage 0–1 stochastic programs. Using this approach, we are able to prove optimality for all previously unsolved instances of a standard test set. Additionally, we implement this scheme as a preprocessing step for PySP, a publicly available and widely used implementation of progressive hedging, and compare this grouping approach with standard grouping approaches on large-scale stochastic unit commitment instances. Finally, the idea is extended to propose a finitely convergent algorithm for two-stage stochastic programs with a finite feasible region.

Soft clustering-based scenario bundling for a progressive hedging heuristic in stochastic service network design

“…4 reduces the number of subproblems from eight to three, with the number of NACs decreasing from seven to two. The benefits of bundle-wise decomposition usually far outweigh the increased difficulty of solving each subproblem (Escudero et al, 2013;Crainic et al, 2014;Ryan et al, 2016), provided that multi-scenario subproblems remain readily manageable.…”

“…However, the scenario-to-bundle assignment was random in nature since no criterion concerning which scenarios would be grouped into the same bundle was given. Ryan et al (2016) formulated the problem of determining the scenario-to-bundle assignment for two-stage stochastic mixed 0-1 programs as a mixed integer program (MIP), the objective function of which is to maximize the bound improvement. Their MIP formulation assumed that the intersection of any two scenario bundles was an empty set and the size of individual bundles was far less than the total number of scenarios.…”

Parallelizing Subgradient Methods for the Lagrangian Dual in Stochastic Mixed-Integer Programming

“…The concept of partitioning has been used in the context of SMIPs in a completely different manner. Instead of completely decomposing the problem so that each scenario can be treated as a separate subproblem, scenarios can be grouped together to form larger subproblems (Boland et al 2016, Maggioni and Pflug 2016, Ryan et al 2016, Sandikçi and Ozaltin 2017. This approach can potentially yield better relaxations at the cost of having more expensive subproblems.…”