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

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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.

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Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, Technion–Israel Institute of Technology, Haifa, 32000, Israel

    A. Beck

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  1. A. Beck
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Correspondence to A. Beck.

Additional information

Communicated B.T. Polyak.

This research was partially supported by the Israel Science Foundation under Grant ISF 489/06.

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Beck, A. Convexity Properties Associated with Nonconvex Quadratic Matrix Functions and Applications to Quadratic Programming. J Optim Theory Appl 142, 1–29 (2009). https://doi.org/10.1007/s10957-009-9539-y

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