scholarly journals SQP alternating direction method with a new optimal step size for solving variational inequality problems with separable structure

2018 ◽  
Vol 12 (1) ◽  
pp. 224-243 ◽  
Author(s):  
Abdellah Bnouhachem ◽  
Themistocles Rassias
Keyword(s):  
Variational Inequality ◽  
Alternating Direction Method ◽  
Convex Programming Problem ◽  
Descent Direction ◽  
Computational Results ◽  
Step Size ◽  
Sqp Method ◽  
Separable Structure ◽  
Alternating Direction ◽  
Optimal Step Size

In this paper, we suggest and analyze a new alternating direction scheme for the separable constrained convex programming problem. The theme of this paper is twofold. First, we consider the square-quadratic proximal (SQP) method. Next, by combining the alternating direction method with SQP method, we propose a descent SQP alternating direction method by using the same descent direction as in [6] with a new step size ?k. Under appropriate conditions, the global convergence of the proposed method is proved. We show the O(1/t) convergence rate for the SQP alternating direction method. Some preliminary computational results are given to illustrate the efficiency of the proposed method.

scholarly journals On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Caihua Chen ◽  
Yuan Shen ◽  
Yanfei You
Keyword(s):  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Objective Functions ◽  
Step Size ◽  
Mild Conditions ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
Optimal Step Size ◽  
Linearly Constrained ◽  
Kkt Points

We consider a class of linearly constrained separable convex programming problems whose objective functions are the sum of three convex functions without coupled variables. For those problems, Han and Yuan (2012) have shown that the sequence generated by the alternating direction method of multipliers (ADMM) with three blocks converges globally to their KKT points under some technical conditions. In this paper, a new proof of this result is found under new conditions which are much weaker than Han and Yuan’s assumptions. Moreover, in order to accelerate the ADMM with three blocks, we also propose a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions.

scholarly journals A Simple Alternating Direction Method for the Conic Trust Region Subproblem

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Honglan Zhu ◽  
Qin Ni
Keyword(s):  
Trust Region ◽  
Alternating Direction Method ◽  
New Method ◽  
Quadratic Model ◽  
Descent Direction ◽  
Orthogonal Direction ◽  
Trust Region Subproblem ◽  
One Dimensional ◽  
Alternating Direction ◽  
Low Dimensional

A simple alternating direction method is used to solve the conic trust region subproblem of unconstrained optimization. By use of the new method, the subproblem is solved by two steps in a descent direction and its orthogonal direction, the original conic trust domain subproblem into a one-dimensional subproblem and a low-dimensional quadratic model subproblem, both of which are very easy to solve. Then the global convergence of the method under some reasonable conditions is established. Numerical experiment shows that the new method seems simple and effective.

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