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Optimality Conditions and Semidefinite Linear Programming Duals for Two-Stage Adjustable Robust Quadratic Optimization

Author

Listed:
  • Huan Zhang

    (Chongqing Technology and Business University
    Chongqing University)

  • Xiangkai Sun

    (Chongqing Technology and Business University)

  • Kok Lay Teo

    (School of Mathematical Sciences, Sunway University)

Abstract

This paper is devoted to the study of a class of two-stage adjustable robust quadratic optimization problems with affine decision rules, where both the objective and constraint functions involve spectrahedral uncertain data. By using a new robust constraint qualification condition, necessary and sufficient conditions expressed in terms of linear matrix inequalities are established for the optimal solutions of the adjustable robust quadratic optimization problem. Subsequently, based on the obtained optimality conditions, a semidefinite linear programming (SDP) dual problem of this adjustable robust quadratic optimization problem is proposed. Then, weak and strong duality properties between them are established. The obtained results provide us with a way to find an optimal value of a two-stage adjustable robust quadratic optimization problem by solving its SDP linear dual problem. Furthermore, as a special case, the second-order cone programming (SOCP) dual for the two-stage adjustable robust separable quadratic optimization is considered.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02725-1
    DOI: 10.1007/s10957-025-02725-1
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    References listed on IDEAS

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    1. Xiao-Bing Li & Suliman Al-Homidan & Qamrul Hasan Ansari & Jen-Chih Yao, 2020. "Robust Farkas-Minkowski Constraint Qualification for Convex Inequality System Under Data Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 785-802, June.
    2. Xiangkai Sun & Jiayi Huang & Kok Lay Teo, 2024. "On semidefinite programming relaxations for a class of robust SOS-convex polynomial optimization problems," Journal of Global Optimization, Springer, vol. 88(3), pages 755-776, March.
    3. T. D. Chuong & V. H. Mak-Hau & J. Yearwood & R. Dazeley & M.-T. Nguyen & T. Cao, 2022. "Robust Pareto solutions for convex quadratic multiobjective optimization problems under data uncertainty," Annals of Operations Research, Springer, vol. 319(2), pages 1533-1564, December.
    4. Jiawei Chen & Elisabeth Köbis & Jen-Chih Yao, 2019. "Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 411-436, May.
    5. 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.
    6. A. Takeda & S. Taguchi & R. H. Tütüncü, 2008. "Adjustable Robust Optimization Models for a Nonlinear Two-Period System," Journal of Optimization Theory and Applications, Springer, vol. 136(2), pages 275-295, February.
    7. Gorissen, B.L. & den Hertog, D., 2012. "Approximating the Pareto Set of Multiobjective Linear Programs via Robust Optimization," Other publications TiSEM 666c5307-4a4e-4be4-a0d0-b, Tilburg University, School of Economics and Management.
    8. La Huang & Danyang Liu & Yaping Fang, 2023. "Convergence of an SDP hierarchy and optimality of robust convex polynomial optimization problems," Annals of Operations Research, Springer, vol. 320(1), pages 33-59, January.
    9. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2018. "Characterizations for Optimality Conditions of General Robust Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 835-856, June.
    10. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    11. Gorissen, B.L. & den Hertog, D., 2012. "Approximating the Pareto Set of Multiobjective Linear Programs via Robust Optimization," Discussion Paper 2012-031, Tilburg University, Center for Economic Research.
    12. Jiawei Chen & Jun Li & Xiaobing Li & Yibing Lv & Jen-Chih Yao, 2020. "Radius of Robust Feasibility of System of Convex Inequalities with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 384-399, February.
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