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Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems

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Abstract

Due to their fundamental nature and numerous applications, sphere constrained polynomial optimization problems have received a lot of attention lately. In this paper, we consider three such problems: (i) maximizing a homogeneous polynomial over the sphere; (ii) maximizing a multilinear form over a Cartesian product of spheres; and (iii) maximizing a multiquadratic form over a Cartesian product of spheres. Since these problems are generally intractable, our focus is on designing polynomial-time approximation algorithms for them. By reducing the above problems to that of determining the L 2-diameters of certain convex bodies, we show that they can all be approximated to within a factor of Ω((log n/n)d/2–1) deterministically, where n is the number of variables and d is the degree of the polynomial. This improves upon the currently best known approximation bound of Ω((1/n)d/2–1) in the literature. We believe that our approach will find further applications in the design of approximation algorithms for polynomial optimization problems with provable guarantees.

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Authors and Affiliations

  1. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Anthony Man-Cho So

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  1. Anthony Man-Cho So
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Correspondence to Anthony Man-Cho So.

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This research was supported by the Hong Kong Research Grants Council (RGC) General Research Fund (GRF) Projects CUHK 416908 and CUHK 419409.

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So, A.MC. Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems. Math. Program. 129, 357–382 (2011). https://doi.org/10.1007/s10107-011-0464-0

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  • DOI: https://doi.org/10.1007/s10107-011-0464-0

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Mathematics Subject Classification (2000)