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 multi-output (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 general-purpose global optimization solver called Baron for solving the virtual beamforming design problem.
Highlights
The following Theorem 2 shows that the sequence {Lk} generated by the ECSDR-BB algorithm is a lower bound of the optimal value of problem (CQP) and the solution x∗ returned by the algorithm is an -optimal solution of the problem
These results show that our proposed ECSDR-BB algorithm is very efficient for globally solving the multi-input multi-output (MIMO) detection problem
In this paper, we considered a class of nonconvex complex quadratic programming problems (i.e., problem (CQP)), which finds many important signal processing applications
Summary
We apply our proposed branch-and-bound algorithm to solve three important signal processing problems, i.e., the MIMO detection problem [2], [3], the unimodular radar code design problem [4]–[6], and the virtual beamforming design problem [7]. The SNR of the considered radar system can be expressed as c · x† M−1 pp† ∗ x, where c is a constant (depending only on the cases of the nonfluctuaing and fluctuating target), x ∈ Cn is the unimodular radar code to be designed, M is the positive definite covariance matrix of some unknown zero-mean complex Gaussian noise vector, p = [1, ei2π fd Tr , . A. CONVENTIONAL SDR By introducing an n × n complex matrix X = xx†, problem (CQP) can be equivalently reformulated as min 1 Q X + Re c†x x,X 2 s.t. X = xx†, where Xii is the i-th diagonal entry of X. There is a nonzero gap between problem (P) and its relaxation (CSDR), where the gap between two problems in this paper refers to the absolute value of the difference between the optimal values of the two problems
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