AN APPROXIMATE PROXIMAL POINT ALGORITHM FOR NONLINEAR COMPLEMENTARITY PROBLEMS

Abstract

In this paper, we suggest and analyze a new method for solving nonlinear complementarity problems (NCP) where the underlying function F is co-coercive. The theme of this paper is twofold. First, we consider the logarithmic-quadratic proximal (LQP) method which was introduced by Auslender, Teboulle and Ben-Tiba (A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12, 31–40, 1999). Next, we propose a new modified LQP method by using a new direction with a new step size αk. We show that the method is
globally convergent. Some preliminary computational results are given to illustrate the efficiency of the proposed method.

Keywords

• Nonlinear complementarity problems,Co-coercive operator,Logarithmicquadratic proximal method,2000 AMS Classification: 49 J 40

AN APPROXIMATE PROXIMAL POINT ALGORITHM FOR NONLINEAR COMPLEMENTARITY PROBLEMS

Abstract

References

1. Auslender, A. Optimization M´ethodes Num´eriques(Mason, Paris, 1976).
2. Auslender, A. and Teboule, M. Interior projection-like methods for monotone variational inequalities, Math. Prog., Ser. A 104, 39–68, 2005.
3. Auslender, A., Teboulle, M. and Ben-Tiba, S. A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12, 31–40, 1999.
4. Auslender, A., Teboulle, M. and Ben-Tiba, S. Interior proximal and multiplier methods based on second order homogenous kernels, Math. Oper. Res. 24, 646–668, 1999.
5. Bnouhachem, A. An LQP method for pseudomonotone variational inequalities, J. Global Optim. 36 (3), 351–363, 2006.
6. Bnouhachem, A. and Noor, M. A. A new predictor-corrector method for pseudomonotone nonlinear complementarity problems, Inter. J. Compt. Math. 85, 1023–1038, 2008.
7. Bnouhachem, A., Noor, M. A., Khalfaoui, M. and Zhaohan, S. A new logarithmic-quadratic proximal method for nonlinear complementarity problems, Appl. Math. Comput. 215, 695– 706, 2009.
8. Bnouhachem, A. and Noor, M. A. An interior proximal point algorithm for nonlinear com- plementarity problems, Nonlinear Analysis: Hybrid Systems 4 (3), 371–380, 2010.

9. Burachik, R. S. and Iusem, A. N. A generalized proximal point algorithm for the variational inequality problem in a Hilbert Space, SIAM J. Optim. 8, 197–216, 1998.
10. Burachik, R. S. and Svaiter, B. F. A relative error tolerance for a family of generalized proximal point methods, Math. Oper. Res. 26 (4), 816–831, 2001.
11. Cottle, R. W. and Dantzig, G. B. Complemenatriy pivot theory of mathematical program- ming, Linear Algbra Appl. 1, 103–125, 1968.
12. Eckestein, J. Approximate iterations in Bregman-function-based proximal algorithms, Math. Prog. 83, 113–123, 1998.
13. Ferris, M. C. and Pang, J. S. Engineering and economic applications of complementariry problems, SIAM Review. 39, 669–713, 1997.
14. Fischer, A. Solution of monotone complementarity problems with locally Lipschitzian func- tions, Math. Prog. 76, 513–532, 1997.
15. Han, D. A new hybrid generalized proximal point algorithm for variational inequality prob- lems, J. Global Optim. 26, 125–140, 2003.
16. Harker, P. T. and Pang, J. S. A damped-Newton method for the linear complementarity problem(Lectures in Applied Mathematics 26, 1990), 265–284.
17. He, B. S., Liao, L. -Z. and Yuan, X. M. A LQP based interior prediction-correction method for nonlinear complementarity problems, Journal of Computational Mathematics 24 (1), 33–44, 2006.
18. Kaplan, A. and Tichatschke, R. On inexact generalized proximal method with a weakned error tolerance criterion, Optimization 53 (1), 3–17, 2004.
19. Lemke, C. E. Bimatrix equilibrium point and mathematical programming, Management Sci. , 681–689, 1965.
20. Marcotte, P. and Dussault, J. P. A Note on a globally convergent Newton method for solving variational inequalities, Operation Research Letters 6, 35–42, 1987.
21. Martinet, B. Regularization d’inequations variationnelles par approximations successives, Revue Francaise d’Informatique et de Recherche Operationelle 4, 154–159, 1970.
22. Mor´e, J. J. Global methods for nonlinear complementarity problems, Math. Oper. Res. 21, 589–614, 1996.
23. Nagurney, A. and Zhang, D. Projected Dnamical Systems and Variational Inequalities with Applications(Kluwer Academic Publishers, Boston, Dordrecht, London, 1996).
24. Noor, M. A. General variational inequalities, Appl. Math. Letters 1, 119–121, 1988.
25. Noor, M. A. Some developments in general variational inequalities, Appl. Math. Comput. 152, 199–277, 2004.
26. Noor, M. A. and Bnouhachem, B. Modified proximal point methods for nonlinear comple- mentarity problems, J. Comput. Appl. Math. 197, 395–405, 2006.
27. Noor, M. A. Extended general variational inequalities, Appl. Math. Letters 22, 182–185, 2009. [28] Rockafellar, R. T. Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1, 97–116, 1976.
28. Rockafellar, R. T. Monotone operators and the proximal point algorithm, SIAM J. Control. Optim. 14, 877–898, 1976.
29. Solodov, M. V. and Svaiter, B. F. An inexact hybrid generalized proximal point algorithms and some new results on the theory of Bregman functions, Math. Oper. Res. 25, 214–230, 2000. [31] Taji, K., Fukushima, M. and Ibaraki, T. A globally convergent Newton method for solving strongly monotone variational inequalities, Math. Prog. 58, 369–383, 1993.
30. Teboulle, M. Convergence of proximal-like algorithms, SIAM J. Optim. 7, 1069–1083, 1997. [33] Xu, Y., He, B. S. and Yuan, X. A hybrid inexact logarithmic-quadratic proximal method for nolinear complementarity problems, J. Math. Anal. Appl. 322, 276–287, 2006.
31. Yan, X., Han, D. and Sun, W. A self-adaptive projection method with improved step-size for solving variational inequalities, Computers & Mathematics with Applications 55, 819–832, 2008.