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Convergent hierarchy of SDP relaxations for a class of semi-infinite convex polynomial programs and applications

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  • Chuong, T.D.
  • Jeyakumar, V.

Abstract

In this paper we examine semidefinite linear programming approximations to a class of semi-infinite convex polynomial optimization problems, where the index sets are described in terms of convex quadratic inequalities. We present a convergent hierarchy of semidefinite linear programming relaxations under a mild well-posedness assumption. We also provide additional conditions under which the hierarchy exhibits finite convergence. These results are derived by first establishing characterizations of optimality which can equivalently be reformulated as linear matrix inequalities. A separation theorem of convex analysis and a sum-of-squares polynomial representation of positivity of real algebraic geometry together with a special variable transformation play key roles in achieving the results. Finally, as applications, we present convergent relaxations for a broad class of robust convex polynomial optimization problems.

Suggested Citation

  • Chuong, T.D. & Jeyakumar, V., 2017. "Convergent hierarchy of SDP relaxations for a class of semi-infinite convex polynomial programs and applications," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 381-399.
    • Handle: RePEc:eee:apmaco:v:315:y:2017:i:c:p:381-399
    DOI: 10.1016/j.amc.2017.07.076
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    References listed on IDEAS

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    1. Kanzi, N. & Nobakhtian, S., 2010. "Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems," European Journal of Operational Research, Elsevier, vol. 205(2), pages 253-261, September.
    2. Li Wang & Feng Guo, 2014. "Semidefinite relaxations for semi-infinite polynomial programming," Computational Optimization and Applications, Springer, vol. 58(1), pages 133-159, May.
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    Cited by:

    1. Thinh, Vo Duc & Chuong, Thai Doan & Le Hoang Anh, Nguyen, 2023. "Formulas of first-ordered and second-ordered generalization differentials for convex robust systems with applications," Applied Mathematics and Computation, Elsevier, vol. 455(C).
    2. Thai Doan Chuong & José Vicente-Pérez, 2023. "Conic Relaxations with Stable Exactness Conditions for Parametric Robust Convex Polynomial Problems," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 387-410, May.
    3. Thai Doan Chuong, 2022. "Second-order cone programming relaxations for a class of multiobjective convex polynomial problems," Annals of Operations Research, Springer, vol. 311(2), pages 1017-1033, April.
    4. 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.
    5. Cao Thanh Tinh & Thai Doan Chuong, 2022. "Conic Linear Programming Duals for Classes of Quadratic Semi-Infinite Programs with Applications," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 570-596, August.

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