Semidefinite approximation for mixed binary quadratically constrained quadratic programs

TL;DR

This paper introduces semidefinite programming relaxation techniques for mixed binary quadratically constrained quadratic programs, providing approximation bounds and performance analysis for both minimization and maximization models relevant to wireless communications.

Contribution

It develops novel SDP relaxation methods for MBQCQP problems and analyzes their approximation ratios, including bounds for both minimization and maximization models.

Findings

Approximation ratio for minimization model bounded by (Q^2(M-Q+1)+M^2) in real case.

Approximation ratio for minimization model bounded by (M(M-Q+1)) in complex case.

Approximation ratio for maximization model is (/\u2207(M)) and is tight up to a constant.

Abstract

Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. We consider both a minimization and a maximization model of this problem. For the minimization model, the objective is to find a minimum norm vector in $N$-dimensional real or complex Euclidean space, such that $M$ concave quadratic constraints and a cardinality constraint are satisfied with both binary and continuous variables. {\color{blue}By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of the minimization model and its SDP relaxation is upper bounded by $\cO(Q^2(M-Q+1)+M^2)$ in the real case and by $\cO(M(M-Q+1))$ in the complex case.} For the maximization model, the goal is to find a maximum norm vector subject to a set of quadratic constraints and a cardinality constraint with both binary and continuous variables. We show that in this case the approximation ratio is bounded from below by $\cO(\epsilon/\ln(M))$ for both the real and the complex cases. Moreover, this ratio is tight up to a constant factor.