Tightness and Equivalence of Semidefinite Relaxations for MIMO Detection

Ruichen Jiang Department of Electronic Engineering, Tsinghua University, Beijing 100084, China Ya-Feng Liu State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Chenglong Bao Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Bo Jiang School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210023, China

Optimization and Control mathscidoc:2206.27002

2021.9
The multiple-input multiple-output (MIMO) detection problem, a fundamental problem in modern digital communications, is to detect a vector of transmitted symbols from the noisy outputs of a fading MIMO channel. The maximum likelihood detector can be formulated as a complex least-squares problem with discrete variables, which is NP-hard in general. Various semidefinite relaxation (SDR) methods have been proposed in the literature to solve the problem due to their polynomial-time worst-case complexity and good detection error rate performance. In this paper, we consider two popular classes of SDR-based detectors and study the conditions under which the SDRs are tight and the relationship between different SDR models. For the enhanced complex and real SDRs proposed recently by Lu et al., we refine their analysis and derive the necessary and sufficient condition for the complex SDR to be tight, as well as a necessary condition for the real SDR to be tight. In contrast, we also show that another SDR proposed by Mobasher et al. is not tight with high probability under mild conditions. Moreover, we establish a general theorem that shows the equivalence between two subsets of positive semidefinite matrices in different dimensions by exploiting a special "separable" structure in the constraints. Our theorem recovers two existing equivalence results of SDRs defined in different settings and has the potential to find other applications due to its generality.
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@inproceedings{ruichen2021tightness,
  title={Tightness and Equivalence of Semidefinite Relaxations for MIMO Detection},
  author={Ruichen Jiang, Ya-Feng Liu, Chenglong Bao, and Bo Jiang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613165928540757350},
  year={2021},
}
Ruichen Jiang, Ya-Feng Liu, Chenglong Bao, and Bo Jiang. Tightness and Equivalence of Semidefinite Relaxations for MIMO Detection. 2021. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613165928540757350.
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