A symmetric version of the generalized alternating direction method of multipliers for two-block separable convex programming

Jing Liu,Min Sun,Yongrui Duan

doi:10.1186/s13660-017-1405-0

Abstract

This paper introduces a symmetric version of the generalized alternating direction method of multipliers for two-block separable convex programming with linear equality constraints, which inherits the superiorities of the classical alternating direction method of multipliers (ADMM), and which extends the feasible set of the relaxation factor α of the generalized ADMM to the infinite interval [1,+infty). Under the conditions that the objective function is convex and the solution set is nonempty, we establish the convergence results of the proposed method, including the global convergence, the worst-case mathcal{O}(1/k) convergence rate in both the ergodic and the non-ergodic senses, where k denotes the iteration counter. Numerical experiments to decode a sparse signal arising in compressed sensing are included to illustrate the efficiency of the new method.

Highlights

1 Introduction We consider the two-block separable convex programming with linear equality constraints, where the objective function is the sum of two individual functions with decoupled variables: min θ (x ) + θ (x )|A x + A x = b, x ∈ X, x ∈ X, ( )
3 Algorithm and convergence results we first describe the symmetric version of the generalized alternating direction method of multipliers (SGADMM) for VI(W, F, θ ) formally, and we prove its global convergence in a contraction perspective and establish its worst-case O( /k) convergence rate in both the ergodic and the non-ergodic senses step by step, where k denotes the iteration counter
4 Numerical experiments we present some numerical experiments to verify the efficiency of SGADMM for solving compressed sensing

Summary

Introduction

For more new development of the ADMM-type methods, including the convergence rate, acceleration techniques, its generalization for solving multi-block separable convex programming and nonconvex, nonsmooth programming, we refer to [ – ]. In this paper, we are going to propose a new generalized ADMM, whose both sub-problems incorporate the relaxation factor α directly. For any x ∈ Rn and G , the G-norm x G of the vector x is defined as x Gx. The effective domain of a closed proper function f : X → (–∞, +∞] is defined as dom(f ) := {x ∈ X |f (x) < +∞}, and the symbol ri(C) denotes the set of all relative interior points of a given nonempty convex set C.

The mixed variational inequality problem

Three matrices and their properties

Conclusions