Skip to main content
SpringerLink
Log in

Gap Functions and Descent Methods for Minty Variational Inequality

  • Conference paper
Optimization and Control with Applications

Part of the book series: Applied Optimization ((APOP,volume 96))

Abstract

A new class of gap functions associated to the variational inequality introduced by Minty is defined. Descent methods for the minimization of the gap functions are analysed in order to develop exact and inexact line-search algorithms for solving strictly and strongly monotone variational inequalities, respectively.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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

  • Auslender A. (1976), Optimization. Methodes Numeriques, Masson, Paris.

    Google Scholar 

  • Avriel M., Diewert W.E., Schaible S. and Ziemba W.T. (1981), Introduction to concave and generalized concave functions, in “Generalized Concavity in Optimization and Economics”, S. Schaible, W.T. Ziemba (Eds.), pp. 21–50.

    Google Scholar 

  • Bank B., Guddat J., Klatte D., Kummer B. and Tammer K. (1983), Nonlinear Parametric Optimization, Birkhauser Verlag.

    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 

  • Giannessi F. (1998), On Minty variational principle, in “New Trends in Mathematical Programming”, F. Giannessi, S. Komlosi, T. Rapcsak (Eds.), Kluwer Academic Publishers, Dordrecht, Boston, London.

    Google Scholar 

  • Harker P.T., Pang J.S. (1990), Finite-dimensional variational inequalities and nonlinear complementarity problem: a survey of theory, algorithms and applications, Mathematical Programming, Vol. 48, pp. 161–220.

    Article  MathSciNet  MATH  Google Scholar 

  • Hogan W. (1973), Directional derivatives for extremal-value functions with applications to the completely convex case, Operations Research, Vol. 21, N.1, pp.188–209.

    MATH  MathSciNet  Google Scholar 

  • John R. (1998), Variational inequalities and pseudomonotone functions: some characterizations, in ”Generalized Convexity, Generalized Monotonicity”, J.P. Crouzeix, J.E. Martinez-Legaz (Eds.), Kluwer Academic Publishers, Dordrecht, Boston, London.

    Google Scholar 

  • Karamardian S. (1976), An existence theorem for the complementary problem, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454.

    Article  MATH  MathSciNet  Google Scholar 

  • Karamardian S. (1967), Strictly quasi-convex functions and duality in mathematical programming, Journal of Mathematical Analysis and Applications, Vol. 20, pp. 344–358.

    Article  MATH  MathSciNet  Google Scholar 

  • Komlosi S. (1999), On the Stampacchia and Minty variational inequalities, in ”Generalized Convexity and Optimization for Economic and Financial Decisions”, G. Giorgi and F. Rossi (Eds.), Pitagora, Bologna, Italy, pp. 231–260.

    Google Scholar 

  • Lemarechal C., Nemirovskii A. and Nesterov Y. (1995), New variants of bundle methods, Mathematical Programming, Vol. 69, pp. 111–147.

    Article  MathSciNet  Google Scholar 

  • Mastroeni G. (1999), Minimax and extremum problems associated to a variational inequality, Rendiconti del Circolo Matematico di Palermo, Vol. 58, pp. 185–196.

    MATH  MathSciNet  Google Scholar 

  • Mastroeni G. (2000), On Minty vector variational inequality, in “Vector Variational Inequalities and Vector Equilibria. Mathematical Theories”, F. Giannessi (Ed.), Kluwer Academic Publishers, Dordrecht, Boston, London pp. 351–361.

    Google Scholar 

  • Minoux M. (1986), Mathematical Programming, Theory and Algorithms, John Wiley, New York.

    MATH  Google Scholar 

  • Minty G. J. (1962), Monotone (non linear) operators in Hilbert space, Duke Math. Journal, Vol. 29, pp. 341–346.

    Article  MATH  MathSciNet  Google Scholar 

  • Ortega J.M., Rheinboldt W.C. (1970), Iterative solutions of nonlinear equations in several variables, Academic Press, New York.

    Google Scholar 

  • Ponstein J. (1967), Seven kind of convexity, S.I.A.M. Rev., 9, pp. 115–119.

    MATH  MathSciNet  Google Scholar 

  • Thompson W.A., Parke D.W. (1973), Some properties of generalized concave functions, Operations Research, Vol. 21, pp. 305–313.

    Article  MathSciNet  MATH  Google Scholar 

  • Yamashita N., Taji K. and Fukushima M. (1997), Unconstrained optimization reformulations of variational inequality problems, Journal of Optimization Theory and Applications, Vol. 92, pp. 439–456.

    Article  MathSciNet  MATH  Google Scholar 

  • Zangwill W.I. (1969), Nonlinear Programming: a unified approach, Prentice-Hall, Englewood Cliffs, New York.

    MATH  Google Scholar 

  • Zhu D.L., Marcotte P. (1994), An extended descent framework for variational inequalities, Journal of Optimization Theory and Applications, Vol. 80, pp. 349–366.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pisa, Via Buonarroti 2, 56127, Pisa, Italy

    Giandomenico Mastroeni

Authors
  1. Giandomenico Mastroeni
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The Hong Kong Polytechnic University, Hong Kong

    Liqun Qi , Koklay Teo  & Xiaoqi Yang ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science+Business Media, Inc.

About this paper

Cite this paper

Mastroeni, G. (2005). Gap Functions and Descent Methods for Minty Variational Inequality. In: Qi, L., Teo, K., Yang, X. (eds) Optimization and Control with Applications. Applied Optimization, vol 96. Springer, Boston, MA. https://doi.org/10.1007/0-387-24255-4_27

Download citation

Publish with us

Policies and ethics

Search

Navigation