Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

  • Authors:

  • Simai He;Zhening Li;Shuzhong Zhang

  • Affiliations:

  • City University of Hong Kong, Department of Management Sciences, Kowloon Tong, Hong Kong;The Chinese University of Hong Kong, Department of Systems Engineering and Engineering Management, Shatin, Hong Kong;The Chinese University of Hong Kong, Department of Systems Engineering and Engineering Management, Shatin, Hong Kong

  • Venue:
  • Mathematical Programming: Series A and B - 20th International Symposium on Mathematical Programming – ISMP 2009
  • Year:
  • 2010

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Abstract

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.