Robust Convex Quadratically Constrained Quadratic Programming with Mixed-Integer Uncertainty

Can Gokalp (cangokalputexas.edu)
Areesh Mittal (areeshmittalutexas.edu)
Grani A. Hanasusanto (grani.hanasusantoutexas.edu)

Abstract: We study robust convex quadratically constrained quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are amenable to exact copositive programming reformulations of polynomial size. The emerging convex optimization problems are NP-hard but admit a conservative semidefinite programming (SDP) approximation that can be solved efficiently. We prove that this approximation is stronger than the popular approximate $\mathcal S$-lemma method for problem instances with only continuous uncertainty. We also show that all results can be extended to the two-stage robust optimization setting if the problem has complete recourse. We assess the effectiveness of our proposed SDP reformulations and demonstrate their superiority over the state-of-the-art solution schemes on stylized instances of least squares, project management, and multi-item newsvendor problems.

Keywords: robust optimization, quadratic programming, copositive programming, semidefinite programming

Category 1: Robust Optimization

Category 2: Nonlinear Optimization (Quadratic Programming )

Category 3: Linear, Cone and Semidefinite Programming

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Entry Submitted: 06/06/2017
Entry Accepted: 06/06/2017
Entry Last Modified: 07/13/2017

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