Content area
Full Text
J Optim Theory Appl (2011) 151:321337
DOI 10.1007/s10957-011-9876-5
Received: 20 August 2010 / Accepted: 30 May 2011 / Published online: 23 June 2011 Springer Science+Business Media, LLC 2011
Abstract Due to its signicant efciency, the alternating direction method (ADM) has attracted a lot of attention in solving linearly constrained structured convex optimization. In this paper, in order to make implementation of ADM relatively easy, some linearized proximal ADMs are proposed and the associated convergence results of the proposed linearized proximal ADMs are given. Additionally, theoretical analysis shows that the relaxation factor for the linearized proximal ADMs can have the same restriction region as that for the general ADM.
Keywords Structured variational inequality Alternating direction method
Augmented Lagrangian method Proximal point algorithm
1 Introduction
Alternating direction method (ADM) was rst introduced by Peaceman and Rochford [13] and developed by Glowinski and Marrocco [4], and by Gabay and Mercier [5]. The original procedure was applied to the numerical solution of the heat equation and to the iterative solution of the linear systems associated with the Laplace equation. Further contributions to the theory and applications of the ADM appeared in a number of important papers, including [68]. The ADM was shown to be equivalent or closely related to many other algorithms, such as the DouglasRachford splitting method, proximal point method, and others [9]. For instance, Gabay [6] showed that ADM
Communicated by P.M. Pardalos.
M.H. Xu ( )
School of Mathematics and Physics, Changzhou University, Changzhou 213164, P.R. China e-mail: mailto:[email protected]
Web End [email protected]
T. Wu
Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China
A Class of Linearized Proximal Alternating Direction Methods
M.H. Xu T. Wu
322 J Optim Theory Appl (2011) 151:321337
is a special case of the method called the DouglasRachford splitting method for monotone operators, and Eckstein and Bertsekas [9] built the relationship between the DouglasRachford splitting method and the proximal point algorithm.
Because of its high efciency, ADM has attracted a lot of attention of many authors in various areas such as convex programming, variational inequalities, and partial differential equations; see, e.g., [8, 1016]. In particular, some novel and attractive applications of ADM have been discovered very recently [1720]. It is shown that many applications can be found in statistical, machine learning, and...