In this paper, we consider a fundamental problem in modern digital communications known as multi-input multi-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite relaxation (SDR) based algorithms have been proposed to solve the problem in the literature. In this paper, we first show that the conventional SDR is generally not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the level of the noise to be below a certain threshold. The above results have answered an open question posed by So in . Numerical simulation results show that our proposed SDR significantly outperforms the conventional SDR in terms of the relaxation gap.
Cheng Lu, Ya-Feng Liu, Wei-Qiang Zhang, and Shuzhong Zhang, Tightness of a new and enhanced semidefinite relaxation for MIMO detection, July, 2018.