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On Minimizing Some Merit Functions for Nonlinear Complementarity Problems under H-Differentiability

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Abstract

In this paper, we describe the H-differentials of some well known NCP functions and their merit functions. We show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C 1-functions, semismooth-functions, and locally Lipschitzian functions. Illustrations are given to show the usefulness of our results. We present also a result on the global convergence of a derivative-free descent algorithm for solving the nonlinear complementarity problem.

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References

  1. Gowda, M.S., Ravindran, G.: Algebraic univalence theorems for nonsmooth functions. J. Math. Anal. Appl. 252, 917–935 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Gowda, M.S.: A note on H-differentiable functions. Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, Maryland 21250 (November, 1998)

  3. Gowda, M.S.: Inverse and implicit function theorems for H-differentiable and semismooth functions. Optim. Methods Softw. 19, 443–461 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983). Reprinted by SIAM, Philadelphia (1990)

    MATH  Google Scholar 

  5. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 952–972 (1977)

    Article  MathSciNet  Google Scholar 

  6. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  Google Scholar 

  8. Qi, L.: C-differentiability, C-differential operators and generalized Newton methods. Research Report, School of Mathematics, The University of New South Wales, Sydney, New South Wales, Australia (January 1996)

  9. Jeyakumar, V., Luc, D.T.: Approximate Jacobians matrices for nonsmooth continuous maps and C 1-optimization. SIAM J. Control Optim. 36, 1815–1832 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tawhid, M.A., Gowda, M.S.: On two applications of H-differentiability to optimization and complementarity problems. Comput. Optim. Appl. 17, 279–299 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tawhid, M.A.: On the local uniqueness of solutions of variational inequalities under H-differentiability. J. Optim. Theory Appl. 113, 149–164 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Facchinei, F., Kanzow, C.: On unconstrained and constrained stationary points of the implicit Lagrangian. J. Optim. Theory Appl. 92, 99–115 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Facchinei, F., Soares, J.: A new merit function for nonlinear complementarity problems and related algorithm. SIAM J. Optim. 7, 225–247 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fischer, A.: A new constrained optimization reformulation for complementarity problems. J. Optim. Theory Appl. 97, 105–117 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jiang, H.: Unconstrained minimization approaches to nonlinear complementarity problems. J. Glob. Optim. 9, 169–181 (1996)

    Article  MATH  Google Scholar 

  16. Kanzow, C.: Nonlinear complementarity as unconstrained optimization. J. Optim. Theory Appl. 88, 139–155 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mangasarian, O.L., Solodov, M.V.: Nonlinear complementarity as unconstrained and constrained minimization. Math. Program. 62, 277–297 (1993)

    Article  MathSciNet  Google Scholar 

  18. Yamashita, N., Fukushima, M.: On stationary points of the implicit Lagrangian for nonlinear complementarity problems. J. Optim. Theory Appl. 84, 653–663 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, Boston (1992)

    MATH  Google Scholar 

  20. Moré, J.J., Rheinboldt, W.C.: On P- and S- functions and related classes of N-dimensional nonlinear mappings. Linear Algebra Appl. 6, 45–68 (1973)

    Article  MATH  Google Scholar 

  21. Song, Y., Gowda, M.S., Ravindran, G.: On characterizations of P- and P 0-properties in nonsmooth functions. Math. Oper. Res. 25, 400–408 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1998)

    Google Scholar 

  23. Fischer, A.: An NCP-function and its use for the solution of complementarity problems. In: Du, D.Z., Qi, L., Womersley, R.S. (eds.) Recent Advances in Nonsmooth Optimization, pp. 88–105. World Scientific, Singapore (1995)

    Google Scholar 

  24. Geiger, C., Kanzow, C.: On the resolution of monotone complementarity problems. Comput. Optim. 5, 155–173 (1996)

    MATH  MathSciNet  Google Scholar 

  25. Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Program. 76, 513–532 (1997)

    Google Scholar 

  26. Fischer, A., Jeyakumar, V., Luc, D.T.: Solution point characterizations and convergence analysis of a descent algorithm for nonsmooth continuous complementarity problems. J. Optim. Theory Appl. 110, 493–513 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, School of Advanced Technologies and Mathematics, Thompson Rivers University, 900 McGill Road, PO Box 3010, Kamloops, BC V2C 5N3, Canada

    M. A. Tawhid

  2. Faculty of Management, McGill University, 1001 Sherbrooke Street West, Montreal, PQ, H3A 1G5, Canada

    J. L. Goffin

Authors
  1. M. A. Tawhid
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  2. J. L. Goffin
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Correspondence to M. A. Tawhid.

Additional information

Communicated by J.P. Crouzeix.

The first author is deeply indebted to Professor M. Seetharama Gowda for his numerous helpful suggestions and encouragement. Special thanks to Professor J.-P. Crouzeix and an anonymous referees for their constructive suggestions which led to numerous improvements in the paper. The research of the first author was supported in part by the Natural Sciences and Engineering Research Council of Canada and Scholar Activity Grant of Thompson Rivers University. The research of the second author was supported by the Natural Sciences and Engineering Research Council of Canada.

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Tawhid, M.A., Goffin, J.L. On Minimizing Some Merit Functions for Nonlinear Complementarity Problems under H-Differentiability. J Optim Theory Appl 139, 127–140 (2008). https://doi.org/10.1007/s10957-008-9409-z

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