Exact Conic Programming Reformulations of Two-Stage Adjustable Robust Linear Programs with New Quadratic Decision Rules

TL;DR

This paper introduces a new quadratic decision rule for two-stage robust linear optimization, providing exact conic reformulations that enhance computational tractability and improve solution performance over traditional affine rules.

Contribution

It develops a novel quadratic decision rule and demonstrates exact conic reformulations for two-stage robust optimization problems, extending applicability to objective-including adjustable variables.

Findings

QDRs outperform ADRs in worst-case scenarios

Exact SDP and SOCP reformulations are derived

Numerical experiments show improved solutions in lot-sizing problems

Abstract

In this paper we introduce a new parameterized Quadratic Decision Rule (QDR), a generalisation of the commonly employed Affine Decision Rule (ADR), for two-stage linear adjustable robust optimization problems with ellipsoidal uncertainty and show that (affinely parameterized) linear adjustable robust optimization problems with QDRs are numerically tractable by presenting exact semi-definite program (SDP) and second order cone program (SOCP) reformulations. Under these QDRs, we also establish that exact conic program reformulations also hold for two-stage linear ARO problems, containing also adjustable variables in their objective functions. We then show via numerical experiments on lot-sizing problems with uncertain demand that adjustable robust linear optimization problems with QDRs improve upon the ADRs in their performance both in the worst-case sense and after simulated realization of the uncertain demand relative to the true solution.