Open Access

An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities

Fixed Point Theory and Applications20072007:076040

DOI: 10.1155/2007/76040

Received: 16 June 2007

Accepted: 19 September 2007

Published: 4 November 2007

Abstract

A new monotonicity, -monotonicity, is introduced, and the resolvant operator of an -monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.

[123456789101112]

Authors’ Affiliations

(1)
Department of Applied Mathematics, Dalian University of Technology

References

  1. Liu WB, Rubio JE: Optimal shape design for systems governed by variational inequalities—I: existence theory for the elliptic case. Journal of Optimization Theory and Applications 1991,69(2):351–371. 10.1007/BF00940649MATHMathSciNetView ArticleGoogle Scholar
  2. Liu WB, Rubio JE: Optimal shape design for systems governed by variational inequalities—II: existence theory for the evolution case. Journal of Optimization Theory and Applications 1991,69(2):373–396. 10.1007/BF00940650MATHMathSciNetView ArticleGoogle Scholar
  3. Ahmad R, Ansari QH, Irfan SS: Generalized variational inclusions and generalized resolvent equations in Banach spaces. Computers & Mathematics with Applications 2005,49(11–12):1825–1835. 10.1016/j.camwa.2004.10.044MATHMathSciNetView ArticleGoogle Scholar
  4. Ding XP: Perturbed proximal point algorithms for generalized quasivariational inclusions. Journal of Mathematical Analysis and Applications 1997,210(1):88–101. 10.1006/jmaa.1997.5370MATHMathSciNetView ArticleGoogle Scholar
  5. Fang Y-P, Huang N-J: -monotone operator and resolvent operator technique for variational inclusions. Applied Mathematics and Computation 2003,145(2–3):795–803. 10.1016/S0096-3003(03)00275-3MATHMathSciNetView ArticleGoogle Scholar
  6. Hassouni A, Moudafi A: A perturbed algorithm for variational inclusions. Journal of Mathematical Analysis and Applications 1994,185(3):706–712. 10.1006/jmaa.1994.1277MATHMathSciNetView ArticleGoogle Scholar
  7. Lee C-H, Ansari QH, Yao J-C: A perturbed algorithm for strongly nonlinear variational-like inclusions. Bulletin of the Australian Mathematical Society 2000,62(3):417–426. 10.1017/S0004972700018931MATHMathSciNetView ArticleGoogle Scholar
  8. Noor MA: Implicit resolvent dynamical systems for quasi variational inclusions. Journal of Mathematical Analysis and Applications 2002,269(1):216–226. 10.1016/S0022-247X(02)00014-8MATHMathSciNetView ArticleGoogle Scholar
  9. Yuan GX-Z: KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics. Volume 218. Marcel Dekker, New York, NY, USA; 1999:xiv+621.Google Scholar
  10. Facchinei F, Pang J-S: Finite-Dimensional Variational Inequalities and Complementarity Problems. Volume 2. Springer, New York, NY, USA; 2003.Google Scholar
  11. Liu Z, Ume JS, Kang SM: Resolvent equations technique for general variational inclusions. Proceedings of the Japan Academy. Series A 2002,78(10):188–193. 10.3792/pjaa.78.188MATHMathSciNetView ArticleGoogle Scholar
  12. Sun D, Sun J: Semismooth matrix-valued functions. Mathematics of Operations Research 2002,27(1):150–169. 10.1287/moor.27.1.150.342MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Juhe Sun et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.