The alternating direction method of multipliers(ADMM) has been proved to be effective for solving two-block separable convex optimization subject to linear constraints. However, it is not necessarily convergent when it is extended to multiple-block case directly. One remedy could be regrouping multiple-block variables into two groups firstly and then adopting the classic ADMM to the resulting model. The two grouped variables are updated in a Gauss–Seidel scheme, while the variables within each group are updated in a Jacobi scheme, which could make it very suitable for parallel computing. However, a special proximal term is added to each subproblem in order to derive convergence, which may slow down the convergence. In this paper, we propose a partial PPA ADMM, which only requires the proximal term in the subproblems of the first group, i.e., the subproblems in the second group are intact. At the end of each iteration, we need to do an extension on the variables with fixed step size. As the subproblems in the second group are unchanged, the resulting sequence should yield better quality as well as potentially faster convergence. Compared with several state-of-the-art multi-block ADMM, preliminary numerical experiments on solving both synthetic and real problems show that our proposed method is effective and promising.
Tech report 1801, Nanjing University of Finance and Economics, Aug/2018