A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides
We study two-stage adjustable robust linear programming in which the right-hand sides are uncertain and belong to a convex, compact uncertainty set. This problem is NP-hard, and the affine policy is a popular, tractable approximation. We prove that under standard and simple conditions, the two-stage problem can be reformulated as a copositive optimization problem, which in turn leads to a class of tractable, semidefinite-based approximations that are at least as strong as the affine policy. We investigate several examples from the literature demonstrating that our tractable approximations significantly improve the affine policy. In particular, our approach solves exactly in polynomial time a class of instances of increasing size for which the affine policy admits an arbitrarily large gap.
KeywordsTwo-stage adjustable robust optimization Robust optimization Bilinear programming Non-convex quadratic programming Semidefinite programming Copositive programming
The authors would like to thank Qihang Lin for many helpful discussions regarding the affine policy at the beginning of the project and Erick Delage and Amir Ardestani-Jaafari for thoughtful discussions, for relaying the specific parameters of the instance presented in Sect. 5.2, and for pointing out an error in one of our codes. The authors are sincerely grateful to two anonymous referees for their comments and insights that have greatly improved the paper.
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