An extended ADMM for 3-block nonconvex nonseparable problems with applications

We consider a 3-block Alternating Direction Method of Multipliers (ADMM) for solving nonconvex nonseparable problems with linear constraints. Inspired by [37, Sun,Toh and Yang,SIAM Journal on Optimization,25 (2015),pp. 882-915], the proposed ADMM follows the Block Coordinate Descent (BCD) cycle order $ 1\to 3\to 2\to 3 $ to guarantee convergence. We analyze its global convergence based on the Kurdyka-Łojasiewicz property. Also, we discuss two useful extensions of the proposed ADMM with $ 2\to 3\to 1\to 3 $ Gauss-Seidel BCD cycle order, and with adding a proximal term for more generally nonseparable problems, respectively. Moreover, we build a brand new model for nonnegative matrix completion, and report numerical experiments to reveal the efficiency and outperformance of the proposed ADMM.