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Complex polynomial optimization has recently gained more and more attentionin both theory and practice. In this paper, we study the optimization of areal-valued general conjugate complex form over various popular constraint setsincluding the m-th roots of complex unity, the complex unit circle, and thecomplex unit sphere. A real-valued general conjugate complex form is ahomogenous polynomial function of complex variables as well as theirconjugates, and always takes real values. General conjugate form optimizationis a wide class of complex polynomial optimization models, which include manyhomogenous polynomial optimization in the real domain with either discrete orcontinuous variables, and Hermitian quadratic form optimization as well as itshigher degree extensions. All the problems under consideration are NP-hard ingeneral and we focus on polynomial-time approximation algorithms withworst-case performance ratios. These approximation ratios improve previousresults when restricting our problems to some special classes of complexpolynomial optimization, and improve or equate previous results whenrestricting our problems to some special classes of polynomial optimization inthe real domain. These algorithms are based on tensor relaxation and randomsampling. Our novel technical contributions are to establish the first set ofprobability lower bounds for random sampling over the m-th root of unity, thecomplex unit circle, and the complex unit sphere, and propose the firstpolarization formula linking general conjugate forms and complex multilinearforms.

Approximation Algorithms for Optimization of Real-Valued General Conjugate Complex Forms