非線性互補問題函數在非線性互補性問題裡扮演很重要的角色。在本篇文章中，我們討論幾個非線性互補性問題函數，其中包含一般化的 費雪－博美斯特函數以及一般化的 自然殘留函數。我們企圖提出幾個關於這些函數的基本性質。

It is well known that NCP-functions play an important role in nonlinear complementarity problem (NCP). In this paper, we study several NCP-functions including the generalized Fisher-Burmeister function, and the generalized Natural-Residual function. We attempt to give
some basic properties for these NCP-functions.

[1] J.-S. Chen, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, Journal of Global Optimization,
36(2006), 565-580.

[2] J. S. Chen, On some NCP-functions based on the generalized Fischer-Burmeister function, Asia-Pacic Journal of Operational Research, 24(2007), 401-420.12

[3] J.-S. Chen, H.-T. Gao and S. Pan, A R-linearly convergent derivative-free algorithm for the NCPs based on the generalized Fischer-Burmeister merit function,
Journal of Computational and Applied Mathematics, 232(2009), 455-471.

[4] J.-S. Chen, Z.-H. Huang, and C.-Y. She, A new class of penalized NCP-functions and its properties, Computational Optimization and Applications, 50(2011),
49-73.

[5] J.-S. Chen, C.-H. Ko, and S.-H. Pan, A neural network based on generalized Fischer-Burmeister function for nonlinear complementarity problems, Information
Sciences, 180(2010), 697-711.

[6] J.-S. Chen, C.-H. Ko, and X.-R. Wu, What is the generalization of natural residual function for NCP, to appear in Pacic Journal of Optimization, January,
2016.

[7] J.-S. Chen and S. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications,
40(2008), 389-404.

[8] J.-S. Chen, S.-H. Pan, and T.-C. Lin, A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs, Nonlinear Analysis: Theory,
Methods and Applications, 72(2010), 3739-3758.

[9] J.-S. Chen, S.-H. Pan, and C.-Y. Yang, Numerical comparison of two eective methods for mixed complementarity problems, Journal of Computational and Applied Mathematics, 234(2010), 667-683.

[10] J.-S. Chen, D. F. Sun, and J. Sun, The SC1 property of the squared norm of the SOC Fischer-Burmeister function, Operations Research Letters, vol. 36(2008),
385-392.

[11] Peng-Fei Ma, Jein-Shan Chen, Chien-Hao Huang, Chun-Hsu Ko, Discovery of new complementarity functions for NCP and SOCCP

[12] Yu-Lin Chang, Jein-Shan Chen, Ching-Yu Yang, Symmetrization of generalized natural residual function for NCP, Operations Research Letters, vol. 43(2015),
354-358.

[13] 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.

[14] A. Fischer, A special Newton-type optimization methods, Optimization, 24(1992),269-284.13

[15] A. Fischer, Solution of the monotone complementarity problem with locally Lipschitzian functions, Mathematical Programming, 76(1997), 513-532.

[16] A. Galantai, Properties and construction of NCP functions, Computational Optimization and Applications, 52(2012), 805-824.

[17] C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems, Computational Optimization and Applications, 5(1996), 155-173.

[18] P. T. Harker and J.-S. Pang, Finite dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications,
Mathematical Programming, 48(1990), 161-220.

[19] C. Kanzow, Nonlinear complementarity as nconstrained optimization, Journal of Optimization Theory and Applications, 88(1996), 139-155.

[20] C. Kanzow, N. Yamashita, and M. Fukushima, New NCP-functions and their properties, Journal of Optimization Theory and Applications, 94(1997), 115-135.

[21] O. L. Mangasarian, Equivalence of the Complementarity Problem to a System of Nonlinear Equations, SIAM Journal on Applied Mathematics, 31(1976), 89-92.

[22] J.-S. Pang, Newton's Method for B-dierentiable Equations, Mathematics of Operations Research, 15(1990), 311-341.

[23] D. Sun and L. Qi, On NCP-functions, Computational Optimization and Applications, 13(1999), 201-220.

[24] H.-Y. Tsai and J.-S. Chen, Geometric views of the generalized Fischer-Burmeister function and its induced merit function, Applied Mathematics and Computation, 237(2014), 31-59, 2014.

[25] N. Yamashita and M. Fukushima, On stationary points of the implict Lagrangian for nonlinear complementarity problems, Journal of Optimization Theory
and Applications, 84(1995), 653-663.

[26] Ferris, MC and J-S Pang, Engineering and economic applications of complementarity problems, SIAM Review, 39(1997), 669-713.

[27] J-S. Pang, Complementarity problems.Handbook of Global Optimization, R Horst and P Pardalos (eds.), MA: Kluwer Academic Publishers, 271-338, 1994.

[28] J.-S. Chen, S.-H. Pan, and T.-C. Lin, A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs (eds.), Nonlinear Analysis:Theory, Methods and Applications, vol. 72, no. 9-10, pp. 3739-3758, 2010.