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Adaptive Middle Proximal ADMM for Multi-block Separable Convex Programming

Jianchao Bai(bjc1987***at***163.com)

Abstract: In the literature, there are few researches on investigating such a typical Alternating Direction Method of Multipliers (ADMM) whose framework includes adaptive techniques and some proximal terms contained only in the middle subproblems. To fill this vacancy, an adaptive middle proximal ADMM is developed to solve the multi-block separable convex optimization problem subject to linearly equality constraints, where the proximal terms are designed only into the middle subproblems and the proximal parameter is restricted into $\left(\frac{3N-8+2(N-3)\delta}{4}, +\infty\right)$ in which $\delta\in [0,1]$ is an adaptive parameter and $N\geq 3$ denotes the number of variables. We analyze that the proposed algorithm can become a Gauss-Seidel scheme when taking $\delta=0$ and Jacobian scheme when forcing $\delta=1.$ The region of the proximal parameter in Gauss-Seidel scheme is equal to that in Jacobian scheme if $N=3$ but is wider than the latter if $N>3$, while the latter can enjoy parallel techniques to deal with some cases involving big data. By means of the variational inequality, a key lemma is provided based on the first-order optimality conditions of each involved subproblem. Then, we use it to show the global convergence and the worst-case ergodic convergence rate measured by the iteration complexity of the proposed algorithm. Compared with several existing algorithms, numerical performance of our algorithm is demonstrated by testing a latent variable Gaussian graphical model selection problem arising in statistical learning.

Keywords: Alternating direction method of multipliers; Adaptive update; Proximal term; Variational inequality; Global convergence

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )


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Entry Submitted: 11/09/2017
Entry Accepted: 11/09/2017
Entry Last Modified: 11/09/2017

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