Sampling Scenario Set Partition Dual Bounds for Multistage Stochastic Programs

Abstract

We consider multistage stochastic programming problems in which the random parameters have finite support, leading to optimization over a finite scenario set. There has been recent interest in dual bounds for such problems, of two types. One, known as expected group subproblem objective (EGSO) bounds, require solution of a group subproblem, which optimizes over a subset of the scenarios, for all subsets of the scenario set that have a given cardinality. Increasing the subset cardinality in the group subproblem improves bound quality, (EGSO bounds form a hierarchy), but the number of group subproblems required to compute the bound increases very rapidly. Another is based on partitions of the scenario set into subsets. Combining the values of the group subproblems for all subsets in a partition yields a partition bound. In this paper, we consider partitions into subsets of (nearly) equal cardinality. We show that the expected value of the partition bound over all such partitions also forms a hierarchy. To make use of these bounds in practice, we propose random sampling of partitions and suggest two enhancements to the approach: sampling partitions that align with the multistage scenario tree structure and use of an auxiliary optimization problem to discover new best bounds based on the values of group subproblems already computed. We establish the effectiveness of these ideas with computational experiments on benchmark problems. Finally, we give a heuristic to save computational effort by ceasing computation of a partition partway through if it appears unpromising.

Stage-t scenario dominance for risk-averse multi-stage stochastic mixed-integer programs

“…There has also been a stream of research studying bounds and approximation for riskaverse multi-stage stochastic mixed-integer programming problems (Maggioni and Pflug 2019;Mahmutogulları et al 2018;Sandıkçı and Özaltın 2017;Bakir et al 2020;Maggioni et al 2014;Maggioni and Pflug 2016;Sandıkçı et al 2013). Our focus in this paper is different from those by providing cutting plane algorithms based on the scenario dominance concept to tackle the computational difficulty of solving risk-averse multi-stage stochastic programs.…”

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