## Abstract and Applied Analysis

### A Continuous Trust-Region-Type Method for Solving Nonlinear Semidefinite Complementarity Problem

#### Abstract

We propose a new method to solve nonlinear semidefinite complementarity problem by combining a continuous method and a trust-region-type method. At every iteration, we need to calculate a second-order cone subproblem. We show the well-definedness of the method. The global convergent result is established.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 589731, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606936

Digital Object Identifier
doi:10.1155/2014/589731

Mathematical Reviews number (MathSciNet)
MR3208548

Zentralblatt MATH identifier
07022667

#### Citation

Ji, Ying; Wang, Tienan; Li, Yijun. A Continuous Trust-Region-Type Method for Solving Nonlinear Semidefinite Complementarity Problem. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 589731, 9 pages. doi:10.1155/2014/589731. https://projecteuclid.org/euclid.aaa/1412606936

#### References

• M. S. Gowda and Y. Song, “On semidefinite linear complementarity problems,” Mathematical Programming. A Publication of the Mathematical Programming Society, vol. 88, no. 3, pp. 575–587, 2000.
• M. Shibata, N. Yamashita, and M. Fukushima, “The extended semidefinite linear complementarity problem: a reformulation approach,” in Nonlinear Analysis and Convex Analysis, W. Takahashi and T. Tanaka, Eds., pp. 326–332, World Scientific, Singapore, 1999.
• X. Chen and P. Tseng, “Non-interior continuation methods for solving semidefinite complementarity problems,” Mathematical Programming. A Publication of the Mathematical Programming Society, vol. 95, no. 3, pp. 431–474, 2003.
• P. Tseng, “Merit functions for semi-definite complementarity problems,” Mathematical Programming, vol. 83, no. 2, pp. 159–185, 1998.
• Z. H. Huang and J. Y. Han, “Non-interior continuation method for solving the monotone semidefinite complementarity problem,” Applied Mathematics and Optimization, vol. 47, no. 3, pp. 195–211, 2003.
• N. Yamashita and M. Fukushima, “A new merit function and a descent method for semidefinite complementarity problems,” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, Eds., vol. 22, pp. 405–420, Kluwer Academic, Dordrecht, The Netherlands, 1999.
• S.-J. Qu, M. Goh, and X. Zhang, “A new hybrid method for nonlinear complementarity problems,” Computational Optimization and Applications, vol. 49, no. 3, pp. 493–520, 2011.
• Y. Ji, K.-C. Zhang, S.-J. Qu, and Y. Zhou, “A trust-region method by active-set strategy for general nonlinear optimization,” Computers & Mathematics with Applications, vol. 54, no. 2, pp. 229–241, 2007.
• C. Kanzow, “Some noninterior continuation methods for linear complementarity problems,” SIAM Journal on Matrix Analysis and Applications, vol. 17, no. 4, pp. 851–868, 1996.
• F. Palacios-Gomez, L. Lasdon, and M. Engquist, “Nonlinear optimization by successive linear programming,” Management Science, vol. 28, no. 10, pp. 1106–1120, 1982.
• J. Z. Zhang, N.-H. Kim, and L. Lasdon, “An improved successive linear programming algorithm,” Management Science, vol. 31, no. 10, pp. 1312–1331, 1985.
• C. Kanzow, C. Nagel, H. Kato, and M. Fukushima, “Successive linearization methods for nonlinear semidefinite programs,” Computational Optimization and Applications, vol. 31, no. 3, pp. 251–273, 2005.
• A. Fischer, “Solution of monotone complementarity problems with locally Lipschitzian functions,” Mathematical Programming, vol. 76, no. 3, pp. 513–532, 1997.
• K. C. Toh, R. H. Tutuncij, and M. J. Todd, SDPT3 version 3.02-a MATLAB software for semidefinite-quadratic-linearprogramming, 2002, http://www.math.nus.edu.sg/$\sim\,\!$mattohkc/ sdpt3.html.
• Z. S. Yu, “A descent algorithm for the extended semidefinite linear complementarity problem,” International Journal of Nonlinear Science, vol. 5, no. 1, pp. 89–96, 2008. \endinput