Comparison of two approximal proximal point algorithms for monotone variational inequalities

O221.2; Proximal point algorithms (PPA) are attractive methods for solving monotone variational inequalities (MVI). Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA (APPA)are developed for practical applications. In this pa...

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Vydané v:	Journal of Zhejiang University. A. Science Ročník 8; číslo 6; s. 969 - 977
Hlavný autor:	Tao, Min
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	School of Applied Mathematics and Physics, Nanjing University of Posts and Telecommunications, Nanjing 210003, China 01.05.2007
Predmet:	Projection and contraction methods Proximal point algorithm (PPA) Prediction and correction Monotone varia tional inequality (MVI) Approximate PPA (APPA)
ISSN:	1673-565X, 1862-1775
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Popis
Shrnutí:	O221.2; Proximal point algorithms (PPA) are attractive methods for solving monotone variational inequalities (MVI). Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA (APPA)are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ.For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration.And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm Ⅱ is feasible and easy to implement with relatively low computation load.
ISSN:	1673-565X 1862-1775
DOI:	10.1631/jzus.2007.A0969