On the Optimal Proximal Parameter of an ADMM-like Splitting Method for Separable Convex Programming

He，Bingsheng1,2; Yuan，Xiaoming3

10.1007/978-981-16-2701-9_8

2021

2194-1009

2194-1017

Springer Proceedings in Mathematics and Statistics

360

139-163

We proposed an ADMM-like splitting method in [11] for solving convex minimization problems with linear constraints and multi-block separable objective functions. Its proximal parameter is required to be sufficiently large to theoretically ensure the convergence, despite that a smaller value of this parameter is preferred for numerical acceleration. Empirically, this method has been applied to solve various applications with relaxed restrictions on the parameter, yet no rigorous theory is available for guaranteeing the convergence. In this paper, we identify the optimal (smallest) proximal parameter for this method and clarify some ambiguity in selecting this parameter for implementation. For succinctness, we focus on the case where the objective function is the sum of three functions and show that the optimal proximal parameter is 0.5. This optimal proximal parameter generates positive indefiniteness in the regularization of the subproblems, and thus its convergence analysis is significantly different from those for existing methods of the same kind in the literature, which all require positive definiteness (or positive semi-definiteness plus additional assumptions) of the regularization. We establish the convergence and estimate the convergence rate in terms of iteration complexity for the improved method with the optimal proximal parameter.

Convergence analysis Convex programming Positive indefinite proximal regularization Splitting method

第一

英语

EI

20214110989574

Iterative methods

Numerical Methods:921.6

2-s2.0-85116471245

Scopus

1.Department of Mathematics,Southern University of Science and Technology,Shenzhen,China
2.Department of Mathematics,Nanjing University,Nanjing,China
3.Department of Mathematics,The University of Hong Kong,Pok Fu Lam,Hong Kong

He，Bingsheng,Yuan，Xiaoming. On the Optimal Proximal Parameter of an ADMM-like Splitting Method for Separable Convex Programming[C],2021:139-163.