Approximation methods for complex polynomial optimization

Bo Jiang, Zhening Li, Shuzhong Zhang

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Abstract

Complex polynomial optimization problems arise from real-life applications including radar code design, MIMO beamforming, and quantum mechanics. In this paper, we study complex polynomial optimization models where the objective function takes one of the following three forms: (1) multilinear; (2) homogeneous polynomial; (3) symmetric conjugate form. On the constraint side, the decision variables belong to one of the following three sets: (1) the m-th roots of complex unity; (2) the complex unity; (3) the Euclidean sphere. We first discuss the multilinear objective function. Polynomial-time approximation algorithms are proposed for such problems with assured worst-case performance ratios, which depend only on the dimensions of the model. Then we introduce complex homogeneous polynomial functions and establish key linkages between complex multilinear forms and the complex polynomial functions. Approximation algorithms for the above-mentioned complex polynomial optimization models with worst-case performance ratios are presented. Numerical results are reported to illustrate the effectiveness of the proposed approximation algorithms.
Original languageEnglish
Pages (from-to)219-248
JournalComputational Optimization and Applications
Volume59
Issue number1-2
Early online date18 Feb 2014
DOIs
Publication statusPublished - Oct 2014

Keywords

  • polynomial optimization
  • complex programming
  • complex tensor
  • approximation algorithm
  • tensor relaxation

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