In this thesis, we investigate the curves which are intersected by a vertical plane
and surface based on a certain NCP function. We also study convexity and differentiability of the curves which will help us look into the curves efficiently. In most cases, the inflection points of the curves cannot be figured out exactly. Therefore, we turn to estimate the interval where the curves are convex under this situation. With the help of differentiability and convexity, we look for the local minimum or maximum values of the curves accordingly.

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, no. 2, pp. 115-121, 2020.
3. Y.-L. Chang, J.-S. Chen, and C.-Y. Yang, Symmetrization of generalized natu- ral residual function for NCP, Operations Research Letters, vol. 43, no. 4, pp. 354-358, 2015.
4. J.-S. Chen, On some NCP-functions based on the generalized Fischer-Burmeister function, Asia-Pacific Journal of Operational Research, vol. 24, no. 3, pp. 401-420, 2007.
5. 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.
6. J.-S. Chen, J.-F. Li, and J. Wu, A continuation approach for solving binary quadratic program based on a class of NCP-functions, Applied Mathematics and Com- putation, vol. 219, no. 8, pp. 3975-3992, 2012.
7. A. Galantai, Properties and construction of NCP functions, Computational Opti-
mization and Applications, vol. 52, pp. 805–824, 2012.
8. Hoang Tuy, Convex Analysis and global Optimization, second edition, Springer,
Hanoi, Vietnam, 2016.
9. C.-H. Huang, K. -J. Weng, J. -S. Chen, H. -W. Chu, and M. -Y. Li, On four discrete-type families of NCP-functions, Journal of Nonlinear and Convex Analysis, vol. 20, no. 2, pp. 283-306, 2019.
10. C.-H. Lee, C.-C. Hu, and J.-S. Chen, Using invertible functions to construct
NCP functions, Linear and Nonlinear Analysis, vol. 6, no. 3, pp. 347-369, 2020.
11. P.-F. Ma, J.-S. Chen, C.-H. Huang, and C.-H. Ko, Discovery of new comple- mentarity functions for NCP and SOCCP, Computational and Applied Mathematics, vol. 37, no. 5, pp. 5727-5749, 2018.
12. H.-Y. Tsai and J.-S. Chen, Geometric views of the generalized Fischer- Burmeister function and its induced merit function, Applied Mathematics and Com- putation, vol. 237, June 15, pp. 31-59, 2014.