Skip to main content
SpringerLink
Log in

A class of projection and contraction methods for monotone variational inequalities

  • Published:
Applied Mathematics and Optimization Aims and scope Submit manuscript

Abstract

In this paper we introduce a new class of iterative methods for solving the monotone variational inequalities

$$u* \in \Omega , (u - u*)^T F(u*) \geqslant 0, \forall u \in \Omega .$$

Each iteration of the methods presented consists essentially only of the computation ofF(u), a projection to Ω,v:=P Ω[u-F(u)], and the mappingF(v). The distance of the iterates to the solution set monotonically converges to zero. Both the methods and the convergence proof are quite simple.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bruck, R. E. (1975), An iterative solution of a variational inequality for certain monotone operators in Hilbert space, Bulletin of the American Mathematical Society, 81:890–892.

    Article  MATH  MathSciNet  Google Scholar 

  2. Dafermos, S. (1983), An iterative scheme for variational inequalities, Mathematical Programming, 26:40–47.

    Article  MATH  MathSciNet  Google Scholar 

  3. Eaves, B. C. (1971), On the basic theorem of complementarity, Mathematical Programming, 1:68–75.

    Article  MATH  MathSciNet  Google Scholar 

  4. Harker, P. T., Pang, J. S. (1990), Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Mathematical Programming, 48:161–220.

    Article  MATH  MathSciNet  Google Scholar 

  5. He, B. S. (1992), A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming, Applied Mathematics and Optimization, 25:247–262.

    Article  MATH  MathSciNet  Google Scholar 

  6. He, B. S. (1994), A new method for a class of linear variational inequalities, Mathematical Programming, 66:137–144.

    Article  MathSciNet  Google Scholar 

  7. He, B. S. (1994), Solving a class of linear projection equations, Numerische Mathematik, 68:71–80.

    Article  MATH  MathSciNet  Google Scholar 

  8. Korpelevich, G. M. (1976), The extragradient method for finding saddle points and other problems, Matekon, 12:747–756.

    MATH  Google Scholar 

  9. Pang, J. S., (1985), Asymmetric variational inequality problems over product sets: applications and iterative methods, Mathematical Programming, 31:206–219.

    Article  MATH  MathSciNet  Google Scholar 

  10. Pang, J. S. (1986), Inexact Newton methods for the nonlinear complementarity problem, Mathematical Programming, 36:54–71.

    Article  MATH  MathSciNet  Google Scholar 

  11. Pang, J. S., Chan, D. (1982), Iterative methods for variational and complementarity problems, Mathematical Programming, 24:284–313.

    Article  MATH  MathSciNet  Google Scholar 

  12. Sun, D. F. (1994), A projection and contraction method for the nonlinear complementarity problem and its extensions, Mathematica Numerica Sinica, 16:183–194.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, 210008, Nanjing, People’s Republic of China

    Bingsheng He

Authors
  1. Bingsheng He
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research is supported by the National Natural Science Foundation of the People’s Republic of China and the Natural Science Foundation of Province Jiangsu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, B. A class of projection and contraction methods for monotone variational inequalities. Appl Math Optim 35, 69–76 (1997). https://doi.org/10.1007/BF02683320

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02683320

Key Words

AMS Classification

Search

Navigation