Improved Decision Rule Approximations for Multi-Stage Robust Optimization via Copositive Programming

TL;DR

This paper introduces improved decision rule approximations for multi-stage robust linear optimization problems using copositive programming, offering tighter bounds and enhanced schemes validated by numerical experiments.

Contribution

It develops novel copositive programming reformulations for decision rule approximations, including piecewise schemes, and proves their superiority over existing methods.

Findings

Proposed copositive reformulations yield tighter approximations.

New piecewise decision rules improve solution quality.

Numerical experiments confirm the superiority of the methods.

Abstract

We study decision rule approximations for generic multi-stage robust linear optimization problems. We consider linear decision rules for the case when the objective coefficients, the recourse matrices, and the right-hand sides are uncertain, and consider quadratic decision rules for the case when only the right-hand sides are uncertain. The resulting optimization problems are NP-hard but amenable to copositive programming reformulations that give rise to tight conservative approximations. We further enhance these approximations through new piecewise decision rule schemes. Finally, we prove that our proposed approximations are tighter than the state-of-the-art schemes and demonstrate their superiority through numerical experiments.