[Paper Review] Improved Decision Rule Approximations for Multi-Stage Robust Optimization via Copositive Programming

Guanglin Xu, Grani A. Hanasusanto|arXiv (Cornell University)|Aug 19, 2018

Advanced Optimization Algorithms Research5 citations

TL;DR

This paper proposes improved decision rule approximations for multi-stage robust optimization using copositive programming, enabling tighter and tractable solutions for problems with uncertain objective coefficients, recourse matrices, and right-hand sides. By leveraging piecewise linear decision rules and semidefinite programming relaxations, the method achieves superior optimality bounds compared to state-of-the-art approaches while remaining computationally scalable.

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.

Motivation & Objective

• To address the computational intractability of multi-stage robust optimization (MSRO) under general uncertainty structures.
• To develop tighter conservative approximations for MSRO when recourse is not fixed, overcoming limitations of traditional linear decision rules.
• To enhance approximation quality through piecewise linear decision rules with provably tighter bounds than existing schemes.
• To provide a tractable solution framework via semidefinite programming reformulations of copositive programs.

Proposed method

• Formulates a lifted uncertain parameter space using piecewise linear lifting, particularly focusing on the first coordinate axis for structural simplification.
• Applies linear decision rules to the lifted uncertain parameters, transforming the problem into a semi-infinite linear program.
• Reformulates the worst-case subproblem as a completely positive program using the cone of completely positive matrices over a lifted conic set.
• Derives tractable semidefinite relaxations by replacing the completely positive cone with a valid, semidefinite-representable outer approximation.
• Employs conic duality and standard relaxation techniques to obtain finite, solvable reformulations of the worst-case maximization problem.
• Generalizes the approach to multiple coordinate axes, ensuring applicability to broader MSRO formulations.

Experimental results

Research questions

• RQ1 Can piecewise linear decision rules be systematically integrated into multi-stage robust optimization to improve solution quality beyond linear rules?
• RQ2 How can copositive programming be leveraged to derive tighter conservative approximations for MSRO with non-fixed recourse?
• RQ3 What is the relative tightness of the proposed approximation compared to existing schemes like truncated or segregated linear decision rules?
• RQ4 Can the resulting optimization problems be reformulated into tractable semidefinite programs without sacrificing solution quality?
• RQ5 What is the impact of lifting the uncertainty set on the quality of the decision rule approximation?

Key findings

• The proposed piecewise linear decision rule scheme yields tighter approximations than state-of-the-art methods, including truncated and segregated linear rules.
• The method achieves a provable bound improvement through the use of copositive programming reformulations of the worst-case subproblem.
• The resulting semidefinite relaxation provides a valid upper bound on the worst-case objective, with a finite, tractable reformulation via conic duality.
• The outer approximation of the completely positive cone leads to a computationally efficient solution approach that scales better than polynomial decision rule hierarchies.
• Numerical experiments demonstrate the superiority of the proposed method in terms of optimality gap and computational efficiency.
• The approach is generalizable to problems with uncertainty in objective coefficients, recourse matrices, and right-hand sides, not limited to fixed-recourse settings.