A new criterion for an inexact parallel splitting augmented Lagrangian method

Xipeng Kou,Shengjie Li,Xueyong Wang

doi:10.1186/s13660-015-0660-1

Xipeng Kou, Shengjie Li + Show 1 more

Open Access

https://doi.org/10.1186/s13660-015-0660-1

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Abstract

In this paper, we study the computational method for solving the variational inequality problem with the separable structure and linear constraints. We propose a new relaxed inexact criterion and a prediction-correction approach in the inexact splitting parallel augmented Lagrangian methods, which make it easier to solve the resulting subproblems. Under a mild condition, we prove the global convergence and establish a worst-case convergence rate for the new inexact algorithm. Some numerical experiments show the effectiveness and feasibility of the new inexact method.

Highlights

For solving the VIP, Glowinski and Marrocco [ ] first proposed a Douglas-Rachford alternating direction method of multiplies (ADMM), which can decompose the original problems into subproblems with a smaller scale
Motivated and inspired by the inexact criteria in [, ], in this paper, we present a new inexact criterion (see ( ) and ( )) to solve the subproblems under a very relaxed restriction
3 The inexact parallel splitting augmented Lagrangian method (PSALM) with new inexact criterion for MVI we propose the inexact method for solving MVI (W, F)

Summary

Introduction

We consider the following variational inequality problem (VI) with the separable structure:. In [ ], an inexact splitting parallel augmented Lagrangian method (IPSALM) was proposed to solve the subproblems ( ) and ( ) approximately so that their solutions satisfy a certain inexact criterion and are closed form ones. In [ ], Zhang et al proposed another inexact criterion for generating the prediction step, that is, xk – xk, ξxk + A xk – xk These inexact methods have the common feature that the subproblems or relevant problems are solved approximately at each iteration. Throughout the paper, we make the following assumptions: (A ) It has a closed form solution to compute the projection onto the convex sets X and Y under Euclidean norm. For the given wk ∈ W, let wk be generated by the proposed algorithm and the new iterate is updated by correction form I or II with γ >. By introducing the slack variable y ≥ , the traffic equilibrium problem ( ) is equivalent to x – x∗, f x∗ ≥ , ∀x ∈ ,

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Conclusions