Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications

In this paper, we study a class of nonconvex nonsmooth optimization problems with bilinear constraints, which have wide applications in machine learning and signal processing. We propose an algorithm based on the alternating direction method of multipliers, and rigorously analyze its convergence properties (to the set of stationary solutions). To test the performance of the proposed method, we specialize it to the nonnegative matrix factorization problem and certain sparse principal component analysis problem. Extensive experiments on real and synthetic data sets have demonstrated the effectiveness and broad applicability of the proposed methods.