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Generalized S-Lemma and strong duality in nonconvex quadratic programming

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  • H. Tuy
  • H. Tuan

Abstract

On the basis of a new topological minimax theorem, a simple and unified approach is developed to Lagrange duality in nonconvex quadratic programming. Diverse generalizations as well as equivalent forms of the S-Lemma, providing a thorough study of duality for single constrained quadratic optimization, are derived along with new strong duality conditions for multiple constrained quadratic optimization. The results allow many quadratic programs to be solved by solving one or just a few SDP’s (semidefinite programs) of about the same size, rather than solving a sequence, often infinite, of SDP’s or linear programs of a very large size as in most existing methods. Copyright Springer Science+Business Media, LLC. 2013

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  • H. Tuy & H. Tuan, 2013. "Generalized S-Lemma and strong duality in nonconvex quadratic programming," Journal of Global Optimization, Springer, vol. 56(3), pages 1045-1072, July.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:3:p:1045-1072
    DOI: 10.1007/s10898-012-9917-0
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    References listed on IDEAS

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    1. 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.
    2. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
    3. Kojima, Masakazu & Tuncel, Levent, 2002. "On the finite convergence of successive SDP relaxation methods," European Journal of Operational Research, Elsevier, vol. 143(2), pages 325-341, December.
    4. B. T. Polyak, 1998. "Convexity of Quadratic Transformations and Its Use in Control and Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 553-583, 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. Huu-Quang Nguyen & Ruey-Lin Sheu, 2019. "Geometric properties for level sets of quadratic functions," Journal of Global Optimization, Springer, vol. 73(2), pages 349-369, February.
    3. 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.
    4. Y. Shi & H. D. Tuan & H. Tuy & S. Su, 2017. "Global optimization for optimal power flow over transmission networks," Journal of Global Optimization, Springer, vol. 69(3), pages 745-760, November.

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