A Nonmonotone Filter Method for Minimax Problems

Qi  Zhao; Nan  Guo

doi:10.4236/am.2011.211193

Qi Zhao, Nan Guo

Abstract: In this paper, we propose a modified trust-region filter method algorithm for Minimax problems, which based on the framework of SQP-filter method and associated with the technique of nonmonotone method. We use the SQP subproblem to acquire an attempt step, and use the filter to weigh the effect of the attempt step so as to avoid using penalty function. The algorithm uses the Lagrange function as a merit function and the nonmonotone filter to improve the effect of the algorithm. Under some mild conditions, we prove the global convergence.

Keywords: Minimax Problem, Nonmonotone, Global Convergence, Filter Methods

Cite this paper: nullQ. Zhao and N. Guo, "A Nonmonotone Filter Method for Minimax Problems," Applied Mathematics, Vol. 2 No. 11, 2011, pp. 1372-1377. doi: 10.4236/am.2011.211193.

References

[1] C. Charalambous and A. R. Conn, “An Efficient Method to Solve the Minimax Problem Directly,” SIAM Journal on Numerical Analysis, Vol. 15, No. 1, 1978, pp. 162-187. doi:10.1137/0715011

[2] A. Vardi, “New Minmax Algorithm,” Journal of Optimization Theory and Applications, Vol. 75, No. 3, 1992, pp. 613-634. doi:10.1007/BF00940496

[3] Z. B. Zhu, “An Improved SQP Algorithm for Solving Minimax problems,” Applied Mathematics Letters, Vol. 22, No. 1, 2009, pp. 464-469. doi:10.1016/j.aml.2008.06.017

[4] Y. H. Yu and L. Gao, “Nonmonotone Line Search Algorithm for Constrained Minimax Problems,” Journal of Optimization Theory and Applications, Vol. 115, No. 2, 2002, pp. 419-446. doi:10.1023/A:1020896407415

[5] J. L. Zhou and A. L. Tits, “Nonmonotone Line Search Method for Minimax Problems,” Journal of Optimization Theory and Applications, Vol. 76, No. 3, 1993, pp. 455-476. doi:10.1007/BF00939377

[6] Y. Xue, “A SQP Method for Minimax Problems,” in Chinese, Journal of System Science and Math Science, Vol. 22, No. 3, 2002, pp. 355-364.

[7] R. Fletcher and S. Leyffer, “Nonlinear Programming without a Penalty Function,” Mathematical Programming, Vol. 91, No. 2, 2002, pp. 239-269. doi:10.1007/s101070100244

[8] R. Fletcher, N. I. M. Gould, S. Leyffer, P. L. Toint and A. W?chter, “Global Convergence of a Trust-Region SQP Filter Algorithm for General Nonlinear Programming,” SIAM Journal on Optimization, Vol. 13, No. 3, 2002, pp. 635-659. doi:10.1137/S1052623499357258

[9] R. Fletcher, S. Leyffer and P. L. Toint, “On the GlobalConvergence of a Filter-SQP Algorithm,” SIAM Journal on Optimization, Vol. 13, No. 1, 2002, pp. 44-59. doi:10.1137/S105262340038081X

[10] A. W?chter and L. T. Biegler, “Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence,” SIAM Journal on Optimization, Vol. 16, No. 1, 2005, pp. 1-31. doi:10.1137/S1052623403426556

[11] A. W?chter and L. T. Biegler, “Line Search Filter Methods for Nonlinear Programming: Local Convergence,” SIAM Journal on Optimization, Vol. 16, No. 1, 2005, pp. 32-48. doi:10.1137/S1052623403426544

[12] R. Fletcher, S. Leyffer and P. L. Toint, “A Brief History of Filter Methods,” SIAG/OPT Views and News, Vol. 18, No. 1, 2006, pp. 2-12.

[13] L. P. Huang, “A Filter Method for Minimax Problems,” in Chinese, Post Graduate Thesis, Suzhou University, Suzhou, 2009.

[14] S. Ulbrich, “On the Superlinear Convergence of Trust Region SQP-Filter Algorithm,” Mathmatical Programming, Vol. 100, Series B, 2004, pp. 217-245.

[15] K. Su and D. G. Pu, “A Nonmonotone Filter Trust Region Method for Nonlinear Constrained Optimization,” Journal of Computational and Applied Mathematics, Vol. 22, No. 1, 2009, pp. 230-239. doi:10.1016/j.cam.2008.01.013

[16] R. Fletcher, S. Leyffer and C. G. Shen, “Nonmonotone Filter Method for Nonlinear Optimization,” Argonne National Laboratory, Lemont, 2009.