An Enhanced SDR Based Global Algorithm for Nonconvex Complex Quadratic Programs With Signal Processing Applications

Abstract

In this paper, we consider a class of nonconvex complex quadratic programming (CQP) problems, which find a broad spectrum of signal processing applications. By using the polar coordinate representations of the complex variables, we first derive a new enhanced semidefinite relaxation (SDR) for problem (CQP). Based on the newly derived SDR, we further propose an efficient branch-and-bound algorithm for solving problem (CQP). Key features of our proposed algorithm are: (1) it is guaranteed to find the global solution of the problem (within any given error tolerance); (2) it is computationally efficient because it carefully utilizes the special structure of the problem. We apply our proposed algorithm to solve the multi-input multioutput (MIMO) detection problem, the unimodular radar code design problem, and the virtual beamforming design problem. Simulation results show that our proposed enhanced SDR, when applied to the above problems, is generally much tighter than the conventional SDR and our proposed global algorithm can efficiently solve these problems. In particular, our proposed algorithm significantly outperforms the state-of-the-art sphere decode algorithm for solving the MIMO detection problem in the hard cases (where the number of inputs and outputs is equal or the signal-to-noise-ratio is low) and a state-of-the-art generalpurpose global optimization solver called Baron for solving the virtual beamforming design problem.

Passive Beamforming Design for IRS Communication System with Few-Bit ADCs

“…2) CQP: The result obtained by solving SDP may not be optimal if rank(U) is not one. Next we convert the problem P 1 into the form of CQP, and use branch-and-bound algorithm to find an approximately optimal solution [11].…”

New semidefinite relaxations for a class of complex quadratic programming problems

“…The tightness of the improved semidefinite relaxation can be guaranteed under certain conditions. Similar enhanced semidefinite relaxations have also been designed for other types of complex quadratic programming problems that appear in signal processing [10,12].…”

“…In [10] and [12], Lu et al have proposed a method of representing a complex variable by its polar coordinate form to derive tight semidefinite relaxations. The main idea behind the method is that some valid constraints can be easily derived in terms of polar coordinate variables, while it is hard to discover them in the complex coordinate variables.…”

“…The main idea behind the method is that some valid constraints can be easily derived in terms of polar coordinate variables, while it is hard to discover them in the complex coordinate variables. Based on this method, Lu et al have proposed some improved semidefinite relaxations for several classes of complex quadratic optimization problems in [10,12]. However, for problem (CQP) that contains the constraints arg(x i x j † ) ∈ A ij , the previous semidefinite relaxations proposed in [10,12] are not always applicable.…”

Optimal Solutions for Joint Beamforming and Antenna Selection: From Branch and Bound to Graph Neural Imitation Learning

“…Related Works. B&B was proposed for beamforming problems in [25], [26], and antenna selection problems in [27]- [29]. Particularly, the work in [25] considered single group multicast beamforming problem, the work in [28] considered a joint power allocation and antenna selection problem, the work in [27] considered antenna selection-assisted rate maximization in wiretap channels, and [29] considered receive antenna selection for sum rate maximization.…”