Hybrid Strategies using Linear and Piecewise-Linear Decision Rules for Multistage Adaptive Linear Optimization

TL;DR

This paper explores hybrid decision rules combining linear and piecewise-linear approaches for multistage adaptive linear optimization, balancing solution quality and computational efficiency through empirical and computational analysis.

Contribution

It introduces the first computational study of hybrid decision rules, analyzing the impact of uncertainty resolution distribution on solution quality and computational cost.

Findings

Higher early-stage uncertainty resolution improves solution quality.

Non-increasing hybrid rules outperform in early stages.

Linear decision rules can outperform piecewise-linear rules in simulation.

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 decision rules, the major contribution of this paper is a computational exploration of hybrid 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 conduct a comprehensive computational study for non-increasing (i.e., higher uncertainty resolution in early stages) and non-decreasing (i.e., higher uncertainty resolution in late stages) hybrid decision rules to illustrate the trade-off between solution quality and computational cost. We also demonstrate a case where a linear decision rule is superior to a piecewise-linear decision rule within a simulator environment, which supports the need to assess the quality of decision rules obtained from a look-ahead model within a simulator rather than just using the look-ahead model's objective function value.