Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods

Language

English

Abstract

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. To solve the subproblems of these algorithms, the projection method takes the iteration in form of u k+1 = P Ω[u k - α kd k]. Interestingly, many of them can be paired such that ũ k = P Ω[u k - β kF(v k)] = P Ω[ũ k - (d k 2 - Gd k 1)], where inf{β k} > 0 and G is a symmetric positive definite matrix. In other words, this projection equation offers a pair of directions, i.e., d k 1 and d k 2 for each step. In this paper, for various APPAs we present a unified framework involving the above equations. Unified characterization is investigated for the contraction and convergence properties under the framework. This shows some essential views behind various outlooks. To study and pair various APPAs for different types of variational inequalities, we thus construct the above equations in different expressions according to the framework. Based on our constructed frameworks, it is interesting to see that, by choosing one of the directions (d k 1 and d k 2) those studied proximal-like methods always utilize the unit step size namely α k ≡ 1. © Springer Science+Business Media, LLC 2010.

Keywords

Contraction methods, Monotone, Variational inequality

Publication Date

2012

Source Publication Title

Computational Optimization and Applications

Volume

51

Issue

2

Start Page

649

End Page

679

Publisher

Springer Verlag

DOI

10.1007/s10589-010-9372-0

Link to Publisher's Edition

http://dx.doi.org/10.1007/s10589-010-9372-0

ISSN (print)

09266003

ISSN (electronic)

15732894

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