Skip to main content
SpringerLink
Log in

On the Contraction and Nonexpansiveness Properties of the Marginal Mappings in Generalized Variational Inequalities Involving Co-Coercive Operators

  • Conference paper
Generalized Convexity, Generalized Monotonicity and Applications

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 77))

Abstract

We investigate the contraction and nonexpansiveness properties of the marginal mappings for gap functions in generalized variational inequalities dealing with strongly monotone and co-coercive operators in a real Hilbert space. We show that one can choose regularization operators such that the solution of a strongly monotone variational inequality can be obtained as the fixed point of a certain contractive mapping. Moreover a solution of a co-coercive variational inequality can be computed by finding a fixed point of a certain nonexpansive mapping. The results give a further analysis for some methods based on the auxiliary problem principle. They also lead to new algorithms for solving generalized variational inequalities involving co-coercive operators. By the Banach contraction mapping principle the convergence rate can be easily established.

This work was completed during the visit of the second author at the Department of Mathematics, University of Namur (FUNDP), Namur, Belgium

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Anh, P.N. and Muu, L.D. (2002), The Banach Iterative Procedure for Solving Monotone Variational Inequality. Hanoi Institute of Mathematics, Preprint No. 05.

    Google Scholar 

  • Aubin, J.P. and Ekeland, I. (1984), Applied Nonlinear Analysis, Wiley, New York.

    Google Scholar 

  • Auslender, A. (1976), Optimisation: Méthodes Numériques, Masson, Paris.

    Google Scholar 

  • Browder, F.E. (1966) On the Unification of the Calculus of Variations and the Theory of Monotone Nonlinear Operators in Banach Spaces, Proc. Nat. Acad. Sci. USA, Vol. 56, pp. 419–425.

    Article  MATH  MathSciNet  Google Scholar 

  • Browder, F.E. and Petryshyn, W.V. (1967), Construction of Fixed Points of Nonlinear Mappings in Hilbert Space, J. on Mathematical Analysis and Applications, Vol. 20, pp. 197–228.

    Article  MathSciNet  Google Scholar 

  • Clarke, F.H. (1983), Optimization and Nonsmooth Analysis, Wiley, New-Yowk.

    Google Scholar 

  • Cohen, G. (1988), Auxiliary Problem Principle Extended to Variational Inequalities, J. of Optimization Theory and Applications, Vol. 59, pp. 325–333.

    MATH  Google Scholar 

  • Fukushima, M. (1992), Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99–110.

    Article  MATH  MathSciNet  Google Scholar 

  • Goebel, K. and Kirk, W.A. (1990), Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge.

    Google Scholar 

  • Golshtein E.G. and Tretyakov N.V. (1996), Modified Lagrangians and Monotone Maps in Optimization, Wiley, New York.

    Google Scholar 

  • Harker, P.T. and Pang, J.S. (1990), Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: a Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161–220.

    Article  MathSciNet  Google Scholar 

  • Hue, T.T., Strodiot, J.J. and Nguyen, V.H. (2004), Convergence of the Approximate Auxiliary Problem Method for Solving Generalized Variational Inequalities, J. of Optimization Theory and Applications, Vol. 121, pp. 119–145.

    Article  MathSciNet  Google Scholar 

  • Kinderlehrer, D. and Stampacchia, G. (1980), An Introduction to Variational Inequalities and Their Applications, Academic Press, New York.

    Google Scholar 

  • Konnov, I. (2001), Combined Relaxation Methods for Variational Inequalities, Springer, Berlin.

    Google Scholar 

  • Konnov, I. and Kum S. (2001), Descent Methods for Mixed Variational Inequalities in a Hilbert Space, Nonlinear Analysis: Theory, Methods and Applications, Vol. 47, pp. 561–572.

    Article  MathSciNet  Google Scholar 

  • Luo, Z. and Tseng, P. (1991), A Decomposition Property of a Class of Square Matrices, Applied Mathematics, Vol. 4, pp. 67–69.

    MathSciNet  Google Scholar 

  • Marcotte, P. (1995), A New Algorithm for Solving Variational Inequalities, Mathematical Programming, Vol. 33, pp. 339–351.

    Article  MathSciNet  Google Scholar 

  • Marcotte, P. and Wu, J.H. (1995), On the Convergence of Projection Methods: Application to the Decomposition of Affine Variational Inequalities, J. of Optimization Theory and Applications, Vol. 85, pp. 347–362.

    Article  MathSciNet  Google Scholar 

  • Muu, L.D. and Khang, D.B. (1983), Asymptotic Regularity and the Strongly Convergence of the Proximal Point Algorithm, Acta Mathematica Vietnamica, Vol. 8, pp. 3–11.

    Google Scholar 

  • Muu. L.D. (1986), An Augmented Penalty Function Method for Solving a Class of Variational Inequalities, USSR Computational Mathematics and Mathematical Physics, Vol. 12, pp. 1788–1796.

    MATH  Google Scholar 

  • Noor M.A. (1993) General Algorithm for Variational Inequalities, J. Math. Japonica, Vol. 38, pp. 47–53.

    MATH  MathSciNet  Google Scholar 

  • Noor M.A. (2001) Iterative Schemes for Quasimonotone Mixed Variational Inequalities, Optimization, Vol. 50, pp. 29–44.

    MATH  MathSciNet  Google Scholar 

  • Patriksson M, (1997) Merit Functions and Descent Algorithms for a Class of Variational Inequality Problems. Optimization, Vol. 41, pp. 37–55.

    MATH  MathSciNet  Google Scholar 

  • Patriksson M, (1999), Nonlinear Programming and Variational Inequality Problems, Kluwer, Dordrecht.

    Google Scholar 

  • Rockafellar, R.T. (1976), Monotone Operators and the Proximal Point Algorithm, SIAM J. on Control, Vol. 14, pp. 877–899.

    Article  MATH  MathSciNet  Google Scholar 

  • Rockafellar, R.T. (1979), Convex Analysis, Princeton Press, New Jersey.

    Google Scholar 

  • Salmon, G., Nguyen, V.H. and Strodiot, J.J. (2000), Coupling the Auxiliary Problem Principle and Epiconvergence Theory to Solve General Variational Inequalities, J. of Optimization Theory and Applications, Vol. 104, pp. 629–657.

    Article  MathSciNet  Google Scholar 

  • Taji, K. and Fukushima, M. (1996), A New Merit Function and a Successive Quadratic Programming Algorithm for Variational Inequality Problems, SIAM J. on Optimization, Vol. 6, pp. 704–713.

    Article  MathSciNet  Google Scholar 

  • Taji, K., Fukushima, M. and Ibaraki (1993), A Global Convergent Newton Method for Solving Monotone Variational Inequality Problems, Mathematical Programming, Vol. 58, pp. 369–383.

    Article  MathSciNet  Google Scholar 

  • Tseng, P. (1990), Further Applications of Splitting Algorithm to Decomposition Variational Inequalities and Convex Programming, Mathematical Programming, Vol. 48, pp. 249–264.

    Article  MATH  MathSciNet  Google Scholar 

  • Verma, R.U. (2001), Generalized Auxiliary Problem Principle and Solvability of a Class of Nonlinear Variational Inequalities Involving Cocoercive and Co-Lipschitzian Mappings, J. of Inequalities in Pure and Applied Mathematics, Vol. 2, pp. 1–9.

    MATH  Google Scholar 

  • Wu, J.H., Florian, M. and Marcotte, P. (1992), A General Descent Framework for the Monotone Variational Inequality Problem, Mathematical Programming, Vol. 53, pp. 99–110.

    Article  MathSciNet  Google Scholar 

  • Zhu, D. and Marcotte, P. (1994), An Extended Descent Framework for Variational Inequalities, J. of Optimization Theory and Applications, Vol. 80, pp. 349–366.

    Article  MathSciNet  Google Scholar 

  • Zhu, D. and Marcotte, P. (1995), A New Class of Generalized Monotonicity, J. of Optimization Theory and Applications, Vol. 87, pp. 457–471.

    Article  MathSciNet  Google Scholar 

  • Zhu, D. and Marcotte, P. (1996), Co-coercivity and its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities, SIAM J. on Optimization, Vol. 6, pp. 714–726.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Posts and Telecommunications, Institute of Technology, Vietnam

    Pham Ngoc Anh

  2. Hanoi Institute of Mathematics, Vietnam

    Le Dung Muu

  3. Department of Mathematics, University of Namur (FUNDP), Belgium

    Van Hien Nguyen & Jean-Jacques Strodiot

Authors
  1. Pham Ngoc Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Le Dung Muu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Van Hien Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jean-Jacques Strodiot
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RMIT University, Australia

    Andrew Eberhard

  2. University of the Aegean, Greece

    Nicolas Hadjisavvas

  3. University of Avignon, France

    Dinh The Luc

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science + Business Media, Inc.

About this paper

Cite this paper

Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, JJ. (2005). On the Contraction and Nonexpansiveness Properties of the Marginal Mappings in Generalized Variational Inequalities Involving Co-Coercive Operators. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol 77. Springer, Boston, MA. https://doi.org/10.1007/0-387-23639-2_5

Download citation

Publish with us

Policies and ethics

Search

Navigation