Hybrid strategies using linear and piecewise-linear decision rules for multistage adaptive linear optimization

Abstract

Decision rules offer a rich and tractable framework for solving certain classes of multistage adaptive optimization problems. Recent literature has shown the promise of using linear and nonlinear decision rules in which wait-and-see decisions are represented as functions, whose parameters are decision variables to be optimized, of the underlying uncertain parameters. Despite this growing success, solving real-world stochastic optimization problems can become computationally prohibitive when using nonlinear decision rules, and in some cases, linear ones. Consequently, decision rules that offer a competitive trade-off between solution quality and computational time become more attractive. Whereas the extant research has always used homogeneous (i.e., either linear or piecewise-linear) decision rules, the major contribution of this paper is a computational exploration of hybrid decision rules combining the benefits of the two classes of decision rules. We first verify empirically that having higher uncertainty resolution or more linear pieces in early stages is more significant than having it in late stages in terms of solution quality. Then, we compare non-increasing and non-decreasing (i.e., higher uncertainty resolution in early and late stages, respectively) hybrid decision rules in a computational study to illustrate the trade-off between solution quality and computational cost. We also demonstrate a case where, unexpectedly, a linear decision rule is superior to a more complex piecewise-linear decision rule within a simulator. This observation bolsters the need to assess the quality of decision rules obtained from a look-ahead model within a simulator rather than just using the optimal look-ahead objective function value.