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The Proximal Point Algorithm for the P0 Complementarity Problem

  • Nobuo Yamashita
  • Junji Imai
  • Masao Fukushima
Part of the Applied Optimization book series (APOP, volume 50)

Abstract

In this paper we consider a proximal point algorithm (PPA) for solving the nonlinear complementarity problem (NCP) with a P 0 function. PPA was originally proposed by Martinet and further developed by Rockafellar for monotone variational inequalities and monotone operator problems. PPA is known to have nice convergence properties under mild conditions. However, until now, it has been applied mainly to monotone problems. In this paper, we propose a PPA for the NCP involving a P 0 function and establish its global convergence under appropriate conditions by using the Mountain Pass Theorem. Moreover, we give conditions under which it has a superlinear rate of convergence.

Keywords

nonlinear complementarity problem proximal point algorithm P0 function 

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References

  1. [1]
    R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, New York, 1992.zbMATHGoogle Scholar
  2. [2]
    T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming, 75 (1996), 407–439.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    F. Facchinei, Structural and stability properties of P o nonlinear complementarity problems, Mathematics of Operations Research, 23 (1998), 735–745.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    F. Facchinei and C. Kanzow, Beyond monotonicity in regu-larization methods for nonlinear complementarity problems, SIAM Journal on Control and Optimization, 37 (1999), 1150–1161.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM Journal on Optimization, 7 (1997), 225–247.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269–284.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    M.S. Gowda and R. Sznajder, Weak univalence and connectedness of inverse images of continuous functions, Mathematics of Operations Research, 24 (1999), 255–261.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    P.T. Harker and J.S. Pang, Finite dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161–220.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    H. Jiang and L. Qi, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM Journal on Control and Optimization, 35 (1997), 178–193.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    B. Martinet, Regularisation d’inequations variationelles par approximations successives, Revue Française d’Informatique et de Recherche Opérationelle, 4 (1970), 154–159.MathSciNetzbMATHGoogle Scholar
  11. [11]
    R.S. Palais and C.-L. Terng, Critical Point Theory and Sub-manifold Geometry, Lecture Note in Mathematics, 1353, Springer Verlag, Berlin, 1988.Google Scholar
  12. [12]
    J.-S. Pang, Complementarity problems, Handbook of Global Optimization, R. Horst and P. Pardalos (eds.), Kluwer Academic Publishers, Boston, Massachusetts, 1994, pp. 271–338.Google Scholar
  13. [13]
    J.-S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithms, SIAM Journal on Optimization, 3 (1993), 443–465.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    L. Qi, C-differentiability, C-differential operators and generalized Newton methods, Technical Report, School of Mathematics, University of New South Wales, Sydney, Australia, 1996.Google Scholar
  15. [15]
    R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14 (1976), 877–898.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    M.V. Solodov and B.F. Svaiter A globally convergent inexact Newton method for systems of monotone equations, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (eds.), Kluwer Academic Publishers, Boston, Massachusetts, 1998, pp. 355–369.Google Scholar
  17. [17]
    N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Mathematical Programming, 76 (1997), 469–491.MathSciNetzbMATHGoogle Scholar
  18. N. Yamashita and M. Fukushima, The proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem, SIAM Journal on Optimization, to appear.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Nobuo Yamashita
    • 1
  • Junji Imai
    • 1
  • Masao Fukushima
    • 2
  1. 1.Toyota Central R&D Labs., Inc.Nagakute, Nagakute-cho, Aichi-gun, AichiJapan
  2. 2.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan

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