[1] Tao Min,. Comparison of two kinds of approximate proximal point algorithmsfor monotone variational inequalities [J]. Journal of Southeast University (English Edition), 2008, 24 (4): 537-540. [doi:10.3969/j.issn.1003-7985.2008.04.028]

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Comparison of two kinds of approximate proximal point algorithmsfor monotone variational inequalities()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

24

Issue:

2008 4

Page:

537-540

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2008-12-30

Info

Title:

Comparison of two kinds of approximate proximal point algorithmsfor monotone variational inequalities

Author(s):

Tao Min

School of Applied Mathematics and Physics, Nanjing University of Posts and Telecommunications, Nanjing 210046, China

Keywords:

monotone variational inequality;  approximate proximate point algorithm;  inexactness criterion

PACS:

O221.2

DOI:

10.3969/j.issn.1003-7985.2008.04.028

Abstract:

This paper proposes two kinds of approximate proximal point algorithms(APPA)for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter’s APPA in the paper “Error bounds for proximal point subproblems and associated inexact proximal point algorithms” published in 2000. They are both prediction-correction methods which use the same inexactness restriction; the only difference is that they use different search directions in the correction steps. This paper also chooses an optimal step size in the two versions of the APPA to improve the profit at each iteration. Analysis also shows that the two APPAs are globally convergent under appropriate assumptions, and we can expect algorithm 2 to get more progress in every iteration than algorithm 1. Numerical experiments indicate that algorithm 2 is more efficient than algorithm 1 with the same correction step size.

References:

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[2] Bertsekas D P, Tsitsiklis J N.Numerical methods[M].Englewood Cliffs:Prentice-Hall, 1989:267-268.
[3] Rockafellar R T.Augmented Lagrangians and applications of the proximal point algorithm in convex programming[J].Mathematics of Operations Research, 1976, 1:97-116.
[4] Rockafellar R T.Monotone operators and the proximal point algorithm[J].SIAM, Journal on Control and Optimization, 1976, 14:877-898.
[5] Solodov M V, Svaiter B F.Error bounds for proximal point subproblems and associated inexact proximal point algorithms[J].Mathematical Programming:Ser B, 2000, 88:371-389.
[6] Zhu T, Yu Z G.A simple proof for some important properties of the projection mapping[J].Mathematical Inequalities and Applications, 2004, 7:453-456.
[7] Yuan Y X.Numerical linear algebra and optimization[C]//Proceedings of the 2001 International Conference on Numerical Optimization and Numerical Linear Algebra.Beijing:Science Press, 2004:15-41.
[8] He B S, Liao L Z.Improvements of some projection methods for monotone nonlinear variational inequalities [J].J Optim Theory Appl, 2002, 112:111-128.

Memo

Memo:

Biography: Tao Min(1979—), female, master, lecturer, taominnju@yahoo.com.
Citation: Tao Min.Comparison of two kinds of approximate proximal point algorithms for monotone variational inequalities[J].Journal of Southeast University(English Edition), 2008, 24(4):537-540.

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