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Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs

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  • V. Jeyakumar

    (University of New South Wales)

  • J. Vicente-Pérez

    (University of New South Wales)

Abstract

In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard.

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  • V. Jeyakumar & J. Vicente-Pérez, 2014. "Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 735-753, September.
  • Handle: RePEc:spr:joptap:v:162:y:2014:i:3:d:10.1007_s10957-013-0496-0
    DOI: 10.1007/s10957-013-0496-0
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    References listed on IDEAS

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    1. G. Yu, 1998. "Min-Max Optimization of Several Classical Discrete Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 98(1), pages 221-242, July.
    2. H. C. Lai & J. C. Liu & K. Tanaka, 1999. "Duality Without a Constraint Qualification for Minimax Fractional Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(1), pages 109-125, April.
    3. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    4. Schaible, Siegfried & Ibaraki, Toshidide, 1983. "Fractional programming," European Journal of Operational Research, Elsevier, vol. 12(4), pages 325-338, April.
    5. Jeyakumar, V. & Li, G., 2010. "New strong duality results for convex programs with separable constraints," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1203-1209, December.
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    Cited by:

    1. T. D. Chuong & V. Jeyakumar, 2018. "Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers," Journal of Global Optimization, Springer, vol. 72(4), pages 655-678, December.
    2. Thai Doan Chuong & Vaithilingam Jeyakumar, 2020. "Generalized Farkas Lemma with Adjustable Variables and Two-Stage Robust Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 488-519, November.
    3. T. D. Chuong & V. Jeyakumar, 2017. "Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 683-703, May.
    4. Jae Hyoung Lee & Liguo Jiao, 2018. "Solving Fractional Multicriteria Optimization Problems with Sum of Squares Convex Polynomial Data," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 428-455, February.
    5. Vaithilingam Jeyakumar & Gue Myung Lee & Jae Hyoung Lee & Yingkun Huang, 2024. "Sum-of-Squares Relaxations in Robust DC Optimization and Feature Selection," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 308-343, January.
    6. 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.
    7. 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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