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SDP reformulation for robust optimization problems based on nonconvex QP duality

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  • Ryoichi Nishimura
  • Shunsuke Hayashi
  • Masao Fukushima

Abstract

In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvex quadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with “single” (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand’s Lorentz positivity. Copyright Springer Science+Business Media New York 2013

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  • Ryoichi Nishimura & Shunsuke Hayashi & Masao Fukushima, 2013. "SDP reformulation for robust optimization problems based on nonconvex QP duality," Computational Optimization and Applications, Springer, vol. 55(1), pages 21-47, May.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:1:p:21-47
    DOI: 10.1007/s10589-012-9520-9
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    References listed on IDEAS

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    Cited by:

    1. Giovanni P. Crespi & Daishi Kuroiwa & Matteo Rocca, 2017. "Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization," Annals of Operations Research, Springer, vol. 251(1), pages 89-104, April.
    2. Crespi, Giovanni P. & Kuroiwa, Daishi & Rocca, Matteo, 2018. "Robust optimization: Sensitivity to uncertainty in scalar and vector cases, with applications," Operations Research Perspectives, Elsevier, vol. 5(C), pages 113-119.
    3. Matteo Rocca, 2025. "Sensitivity to uncertainty and scalarization in robust multiobjective optimization: an overview with application to mean-variance portfolio optimization," Annals of Operations Research, Springer, vol. 346(2), pages 1671-1686, March.
    4. Huan Zhang & Xiangkai Sun & Kok Lay Teo, 2025. "Optimality Conditions and Semidefinite Linear Programming Duals for Two-Stage Adjustable Robust Quadratic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-23, August.

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