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

Bingsheng He(hebma***at***nju.edu.cn)
Xiaoming Yuan(xmyuan***at***hkbu.edu.hk)

Abstract: An ADMM-based splitting method is proposed in [11] for solving convex minimization problems with linear constraints and multi-block separable objective functions; while a relatively large proximal parameter is required for theoretically ensuring the convergence. In this paper, we further study this method and find its optimal (smallest) proximal parameter. 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 the convergence analysis for the improved method is more challenging than 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.

Keywords: Convex programming, splitting method, positive indefinite proximal regularization, convergence analysis

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

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Entry Submitted: 10/01/2017
Entry Accepted: 10/01/2017
Entry Last Modified: 10/01/2017

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