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Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations

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  • V. Jeyakumar
  • G. Li
  • S. Srisatkunarajah

Abstract

In this paper we present necessary conditions for global optimality for polynomial problems with box or bivalent constraints using separable polynomial relaxations. We achieve this by first deriving a numerically checkable characterization of global optimality for separable polynomial problems with box as well as bivalent constraints. Our necessary optimality conditions can be numerically checked by solving semi-definite programming problems. Then, by employing separable polynomial under-estimators, we establish sufficient conditions for global optimality for classes of polynomial optimization problems with box or bivalent constraints. We construct underestimators using the sum of squares convex (SOS-convex) polynomials of real algebraic geometry. An important feature of SOS-convexity that is generally not shared by the standard convexity is that whether a polynomial is SOS-convex or not can be checked by solving a semidefinite programming problem. We illustrate the versatility of our optimality conditions by simple numerical examples. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • V. Jeyakumar & G. Li & S. Srisatkunarajah, 2014. "Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations," Journal of Global Optimization, Springer, vol. 58(1), pages 31-50, January.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:1:p:31-50
    DOI: 10.1007/s10898-013-0058-x
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    References listed on IDEAS

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    1. Guoyin Li, 2012. "Global Quadratic Minimization over Bivalent Constraints: Necessary and Sufficient Global Optimality Condition," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 710-726, March.
    2. V. Jeyakumar & Guoyin Li, 2011. "Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems," Journal of Global Optimization, Springer, vol. 49(1), pages 1-14, January.
    3. M. Ç. Pinar, 2004. "Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 433-440, August.
    4. 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.
    5. Vaithilingam Jeyakumar & Zhiyou Wu, 2007. "Conditions For Global Optimality Of Quadratic Minimization Problems With Lmi Constraints," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 24(02), pages 149-160.
    6. Yanjun Wang & Zhian Liang, 2010. "Global optimality conditions for cubic minimization problem with box or binary constraints," Journal of Global Optimization, Springer, vol. 47(4), pages 583-595, August.
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    Cited by:

    1. Thai Doan Chuong, 2020. "Semidefinite Program Duals for Separable Polynomial Programs Involving Box Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 289-299, April.
    2. Z. Wu & J. Tian & J. Ugon, 2015. "Global optimality conditions and optimization methods for polynomial programming problems," Journal of Global Optimization, Springer, vol. 62(4), pages 617-641, August.
    3. Xue-Gang Zhou & Xiao-Peng Yang & Bing-Yuan Cao, 2015. "Global optimality conditions for cubic minimization problems with cubic constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(3), pages 243-264, December.

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