# 臺灣博碩士論文加值系統

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 在這篇論文當中，我們觀察出一些互補函數是由可逆函數構造出來，像是有名的費雪-博美斯特函數。根據類似的結構，我們利用可逆函數 e^t 和 ln⁡t發現了其他兩個互補函數。我們還發現了另一種的互補函數，它們是在費雪-博美斯特函數的-a和-b這兩項的前面分別乘上滿足特定條件的連續函數。在第四節中，我們討論這三種互補函數的一般形式，並給一些例子和函數的圖形。一些相關的應用和數值的實驗可以當作是後續研究的主題。
 In this thesis, we observed that some of the NCP functions were constructed by invertible functions. For example, the famous Fischer-Burmeister function was constructed under this presence. According to the similar structure of the Fischer-Burmeister function, we discovered the other two NCP functions accociated with the invertible functions which were e^t and ln⁡t. We also discovered another type of NCP function which was modified by multiplying the continuous function satisfying the required assumptions infront of the terms -a and -b of the Fischer-Burmeister function . In the section 4, we discussed the general format of the three newly discovered NCP functions and gave some examples and graphs of the different types of NCP functions. We leave some other possible applications and numerical tests of those NCP functions as our future reseach topics.
 1 Introduction 12 Preliminaries 43 Three specific NCP functions and their subdifferential 54 Constructions of NCP functions by invertible functions 125 Conclusion 266 References 26
 1. J.H. Alcantara and J.-S. Chen, A novel generalization of the natural residual function and a neural network approach for the NCP, Neurocomputing, vol. 413, pp. 368-382, 2020.2. J.H. Alcantara, C.-H. Lee, C.T. Nguyen, Y.-L. Chang, and J.-S. Chen, On construction of new NCP functions, Operations Research Letters, vol.48, pp.115-121, 2020.3. J.-S. Chen, On some NCP-functions based on the generalized Fischer-Burmeister function, Asia-Pacific Journal of Operational Research, vol. 24, pp. 401-420, 2007.4. J.-S. Chen, C.-H. Ko, and X.-R. Wu, What is the generalization of natural residual function for NCP, Pacific Journal of Optimization, vol. 12, pp. 19-27, 2016.5. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.6. F. Facchinei and J.-S. Pang, Finite-dimensional variational inequality and complementarity problems, Springer-Verlag (New York), vol. 1, pp. 1-10, 2003.7. M. C. Ferris and J.-S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, vol. 39, pp. 669-713, 1997.8. A. Fischer, A special Newton-type optimization methods, Optimization, vol.24, pp. 269-284, 1992.9. A. Galantai, Properties and construction of NCP functions, Computational Optimization and Applications, vol. 52, pp. 805-824, 2012.10. C. Kanzow, Nonlinear complementarity as unconstrained optimization, Journal of Optimization Theory and Applications, vol. 88, pp 139-155, 1996.11. C. Kanzow, N. Yamashita, and M. Fukushima, New NCP-Functions and Their Properties, Journal of Optimization Theory and Applications, vol. 94, pp 115-135, 1997.12. C.-H. Lee, C.-C. Hu, and J.-S. Chen, Using invertible functions to construct NCP functions, Linear and Nonlinear Analysis, vol. 6, pp347-369, 2020.13. T.D. Luca, F. Facchinei, and C. Kanzow, Semismooth Equation Approach To The Solution Of Nonlinear Complementarity Problems, Mathematical Programming, vol. 75, pp. 407–439, 1996.14. Z.-Q. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, In: Ferris, M.C., Pang, J.S. (eds.) Complementarity andVariational Problems: State of the Art, pp. 204-225. SIAM, Philadelphia, 1997.15. P.-F. Ma, J.-S. Chen, C.-H. Huang, and C.-H. Ko, Discovery of new complementarity functions for NCP and SOCCP, Computational and Applied Mathematics, vol. 37, pp. 5727-5749, 2018.16. 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. H.-Y. Tsai and J.-S. Chen, Geometric views of the generalized Fischer-Burmeister function and its induced merit function, Applied Mathematics and Computation, vol. 237, pp 31-59, 2014.
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 1 Lagrangian Globalization method together with new NCP-functions for the Nonlinear Complementarity Problem 2 一個連續的方法在費雪-伯麥斯特函數上去解決二進位的二次的規畫問題

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