Conic Programming Reformulations of Two-Stage Distributionally Robust Linear Programs over Wasserstein Balls (1609.07505v3)

Abstract: Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints.

• The paper provides exact copositive programming reformulations for two-stage distributionally robust linear programs over Wasserstein balls.
• These reformulations translate complex robust problems into polynomial-size copositive programs, enhancing computational tractability over approximation methods.
• The developed techniques are also applicable to traditional robust optimization and simplify to linear programs for 1-Wasserstein uncertainty.

Conic Programming Reformulations of Two-Stage Distributionally Robust Linear Programs

The paper under review presents a detailed investigation into the reformulation of two-stage distributionally robust linear programs (DRLPs) over Wasserstein balls. The focus is on providing exact conic programming reformulations for these robust optimization problems, offering significant advancements in efficiency and solution accuracy.

Summary of Contributions

The authors introduce several reformulations of DRLPs using copositive programming. Specifically, the paper addresses scenarios where the ambiguity set describing possible probability distributions is characterized as a 2-Wasserstein ball centered around a discrete distribution. This formulation incorporates the Wasserstein distance as a measure of distributional deviation, which considers both location and spread differences between distributions and provides a robust framework for decision-making under uncertainty.

Key Contributions Include:

1. Copositive Programming Reformulation: The paper proves that two-stage DRLPs with complete recourse can be exactly reformulated as copositive programs of polynomial size. This contrasts sharply with existing methods that typically rely on approximations via decision rules such as piecewise or affine policies.
2. Approximation for Sufficiently Expensive Recourse: When recourse actions are costly, the authors show that the DRLP can be approximated closely by a sequence of copositive programs. By utilizing computationally tractable semidefinite approximations of the copositive cone, the paper develops scalable solution methods that retain high accuracy.
3. 1-Wasserstein Ball Cases: For cases where the ambiguity set is a 1-Wasserstein ball without support constraints, the DRLP simplifies to a linear program. This result highlights a significant reduction in computational complexity under specific conditions.
4. Robust Optimization Reformulation: The approaches are also applicable to traditional robust optimization problems with polyhedral uncertainty sets. This demonstrates flexibility and the broad appeal of the proposed methods across different classes of optimization problems.

Implications

The findings have practical implications for industries where robust decision-making in the face of distributional uncertainty is critical, such as supply chain management and financial services. By providing exact reformulations, the paper aligns with theoretical advances in copositive programming that offer robustness without compromising tractability.

From a theoretical perspective, this work contributes to the domain of distributionally robust optimization by bridging the gap between theory and application. It offers a refined understanding of how exact reformulations can render complex stochastic problems more tractable.

Future Directions

The research opens avenues for future studies in several directions:

• Computational Approaches: While the proposed methods are innovative, the computational burden of handling large-scale copositive programs remains. Future research could explore more efficient algorithmic techniques and the incorporation of machine learning to estimate ambiguity sets dynamically.
• Alternative Metrics: Expanding the framework to incorporate alternative probabilistic metrics beyond the Wasserstein distance could enhance flexibility in various application domains.
• Generalization to Non-Linear Programs: Extending the given results to non-linear decision rules or objectives, possibly through advances in convexification techniques, remains an intriguing opportunity.

In conclusion, the paper formalizes a crucial advancement in distributionally robust optimization by translating complex multi-stage decision problems into solvable conic programs, fostering resilience and sophistication in computational optimization methodologies.