Approximation methods for complex polynomial optimization

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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$m$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>m</mi> </math> </EquationSource> </InlineEquation>-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 homogenous 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. Copyright Springer Science+Business Media New York 2014