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Convexity Properties Associated with Nonconvex Quadratic Matrix Functions and Applications to Quadratic Programming

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  • A. Beck

    (Technion–Israel Institute of Technology)

Abstract

We establish several convexity results which are concerned with nonconvex quadratic matrix (QM) functions: strong duality of quadratic matrix programming problems, convexity of the image of mappings comprised of several QM functions and existence of a corresponding S-lemma. As a consequence of our results, we prove that a class of quadratic problems involving several functions with similar matrix terms has a zero duality gap. We present applications to robust optimization, to solution of linear systems immune to implementation errors and to the problem of computing the Chebyshev center of an intersection of balls.

Suggested Citation

  • A. Beck, 2009. "Convexity Properties Associated with Nonconvex Quadratic Matrix Functions and Applications to Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 1-29, July.
    • Handle: RePEc:spr:joptap:v:142:y:2009:i:1:d:10.1007_s10957-009-9539-y
    DOI: 10.1007/s10957-009-9539-y
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    References listed on IDEAS

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    1. 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.
    2. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    3. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
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    Cited by:

    1. Zhuoyi Xu & Yong Xia & Jiulin Wang, 2021. "Cheaper relaxation and better approximation for multi-ball constrained quadratic optimization and extension," Journal of Global Optimization, Springer, vol. 80(2), pages 341-356, June.

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