在這篇文章中，我們考慮一類罰非線性互補問題函數，它涵蓋了許多已經很有名的非線性互補問題函數。由這類非線性互補問題函數所得到的價值函數會有有界的水準集而且在一些條件下會有誤差界。

In this paper, we consider a class of penalized NCP-functions, which includes several existing well-known NCP-functions as special cases. The merit function induced
by the class of NCP-functions is shown to have bounded level sets and provide error bounds under mild conditions.

[1] J.-S. Chen, H.-T. Gao and S. PAN, A R-linearly convergent derivative-free al-
gorithm for the NCPs based on the generalized Fischer-Burmeister merit function,
submit,2008.
[2] R.W. Cottle, J.-S. Pang and R.-E. Stone, The Linear Complementarity Prob-
lem, Academic Press, New York, 1992.
[3] P. T. Harker and J.-S. Pang, Finite Dimensional Variational Inequality and
Nonlinear Complementarity Problem: A Survey of Theory, Algorithms and Applica-
tions, Mathematical Programming, vol. 48, pp. 161-220, 1990.
[4] J.-S. Chen, The Semismooth-related Properties of a Merit Function and a Descent
Method for the Nonlinear Complementarity Problem, Journal of Global Optimization,
vol. 36, pp. 565-580, 2006.
[5] J.-S. Chen and S. Pan, A Family of NCP-functions and a Descent Method for the
Nonlinear Complementarity Problem, Computational Optimization and Applications,
vol. 40, pp. 389-404, 2008.
[6] Z.H. Huang, The global linear and lacal quadratic convergence of a non-interior
continuation algorithm for the LCP. IMA J. Numer. Anal., vol. 25, pp. 670-684,
2005.
[7] Z.H. Huang and W.Z. Gu, A smoothing-type algorithm for solving linear comple-
mentarity problems with strong convergence properties, Appl. Math. Optim., vol. 57,
pp.17-29, 2008.
[8] Z.H. Huang , L. Qi, and D. Sun, Sub-quadratic convergence of a smoothing
Newton algorithm for the P0-and monotone LCP. Math. Program., vol. 99, pp.423-
441, 2004.
[9] H.Y. Jiang, M. Fukushima, L. Qi etal., A trust region method for solving gen-
eralized complementarity problems. SIAM. J. Optim., vol. 8, pp.140-157, 1998.
[10] C. Kanzow, N. Yamashita, and M. Fukushima, New NCP-functions and their
properties. J. Optim. Theory Appl., vol. 94, pp.115-135, 1997.
[11] J.S. Pang, Complementarity problems. In: Horst, R., Pardalos, P. (eds) Handbook
of Global Optimization. Kluwer Academic Publishers. Boston. Massachusetts. pp.
271-338, 1994.
[12] N. Yamashita and M. Fukushima, On stationary points of the implict La-
grangian for nonlinear complementarity problems. J. Optim. Theory Appl., vol. 84,
pp.653-663, 1995.
[13] N. Yamashita and M. Fukushima, Modied Newton methods for solving a semis-
mooth reformulation of monotone complementarity problems. Math. Program., vol.76,
pp.469-491, 1997.
[14] S.-L. Hu, Z.-H. Huang and J.-S Chen, Properties of a family of generalized
NCP-functions and a derevative free algotithm for complementarity problems. Journal
of Computational and Applied Mathematics, vol. 230, pp. 69-82, 2009.
[15] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and
Complementarity Problems. Springer Verlag, New York, 2003.
[16] O. L. Mangasarian, Equivalence of the Complementarity Problem to a System
of Nonlinear Equations, SIAM Journal on Applied Mathematics, vol. 31, pp. 89-92,
1976.
[17] J.-S. Pang, Newton's Method for B-dierentiable Equations, Mathematics of Op-
erations Research, vol. 15, pp. 311-341, 1990.
[18] S. Dafermos, An Iterative Scheme for Variational Inequalities, Mathematical Pro-
gramming, vol. 26, pp.40-47, 1983.
[19] F. Facchinei and J. Soares, A New Merit Function for Nonlinear Complemen-
tarity Problems and a Related Algorithm, SIAM Journal on Optimization, vol. 7, pp.
225-247, 1997.
[20] A. Fischer, A Special Newton-type Optimization Methods, Optimization, vol. 24,
pp. 269-284, 1992.
[21] C. Geiger and C. Kanzow, On the Resolution of Monotone Complementarity
Problems, Computational Optimization and Applications, vol. 5, pp. 155-173, 1996.
[22] H. Jiang, Unconstrained Minimization Approaches to Nonlinear Complementarity
Problems, Journal of Global Optimization, vol. 9, pp. 169-181, 1996.
[23] C. Kanzow, Nonlinear Complementarity as Unconstrained Optimization, Journal
of Optimization Theory and Applications, vol. 88, pp. 139-155, 1996.
[24] J.-S. Pang and D. Chan, Iterative Methods for Variational and Complemantarity
Problems, Mathematical Programming, vol. 27, pp. 284-313, 1982.
[25] B. Chen, X. Chen, and C. Kanzow, A Penalized Fischer-Burmeister NCP-
function: Theoretical Investigation and Numerical Results, Applied Mathematics Re-
port 97/28, School of Mathematics, the University of New South Wales, Sydney 2052,
Australia, September 1997.