Complex Quadratic Optimization and Semidefinite Programming

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Abstract
In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson [J. Comput. System Sci., 68 (2004), pp. 442–470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an [m2(1 − cos 2π m)/8π]-approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of π/4 ≈ 0.7854, which is better than the ratio of 2/π ≈ 0.6366 for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349. Key words. Hermitian quadratic functions, approximation ratio, randomized algorithms, com-plex semidefinite programming relaxation AMS subject classifications. 90C20, 90C22 DOI. 10.1137/04061341