Learning-Based Branch-and-Bound for Non-Convex Complex Modulus Constrained Problems With Applications in Wireless Communications

We consider a class of non-convex complex modulus constrained problems (CMCPs), which has many important applications in signal processing for wireless communications, including multicast beamforming and multi-input multi-output detection. Due to the non-convex constraints, traditional optimization-based algorithms either obtain sub-optimal solutions or take exponential complexity to reach the optimum. In this paper, we propose a learning based branch-and-bound (LBB) algorithm for solving the considered CMCPs. LBB regards the search procedure of the global optimal argument-cut based branch-and-bound (AC-BB) algorithm as a sequential decision problem in a binary tree and learns the optimal pruning policy via supervised learning. We first propose to apply ensemble learning to train multiple classifiers and then combine them to achieve better performance. To tackle the imbalanced issue, we propose an undersampling-supervised learning method to sample several balanced subsets and train a classifier on each of them. We also show that the computational complexity of LBB is determined by the depth of the binary tree and give its expression in the worst case. Finally, we validate the proposed LBB with two classic problems in wireless communications, i.e., multicast beamforming and MIMO detection. Numerical results show that LBB runs significantly faster than AC-BB while achieving nearly the same optimal performance.