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Abstract and Applied Analysis
Volume 2014, Article ID 436415, 9 pages
http://dx.doi.org/10.1155/2014/436415
Research Article

A QP-Free Algorithm for Finite Minimax Problems

1College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
2School of Mathematics and Information Science, Yulin Normal University, Yulin, Guangxi 537000, China
3Guangxi Economic Management Cadre College, Nanning, Guangxi 530007, China

Received 5 April 2014; Accepted 5 June 2014; Published 24 June 2014

Academic Editor: Chong Li

Copyright © 2014 Daolan Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked 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, pp. 162–187, 1978. View at Google Scholar · View at MathSciNet
  2. S. P. Han, “Variable metric methods for minimizing a class of nondifferentiable functions,” Mathematical Programming, vol. 20, no. 1, pp. 1–13, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  3. M. Gaudioso and M. F. Monaco, “A bundle type approach to the unconstrained minimization of convex nonsmooth functions,” Mathematical Programming, vol. 23, no. 2, pp. 216–226, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  4. W. Hare and J. Nutini, “A derivative-free approximate gradient sampling algorithm for finite minimax problems,” Computational Optimization and Applications, vol. 56, no. 1, pp. 1–38, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  5. R. B. Kearfott, S. Muniswamy, Y. Wang, X. Li, and Q. Wang, “On smooth reformulations and direct non-smooth computations for minimax problems,” Journal of Global Optimization, vol. 57, no. 4, pp. 1091–1111, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  6. F. G. Vazquez, H. Günzel, and H. Th. Jongen, “On logarithmic smoothing of the maximum function,” Annals of Operations Research, vol. 101, pp. 209–220, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. Xu, “Smoothing method for minimax problems,” Computational Optimization and Applications, vol. 20, no. 3, pp. 267–279, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  8. E. Polak, J. O. Royset, and R. S. Womersley, “Algorithms with adaptive smoothing for finite minimax problems,” Journal of Optimization Theory and Applications, vol. 119, no. 3, pp. 459–484, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. Li and J. Huo, “Inexact smoothing method for large scale minimax optimization,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2750–2760, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  10. E. Y. Pee and J. O. Royset, “On solving large-scale finite minimax problems using exponential smoothing,” Journal of Optimization Theory and Applications, vol. 148, no. 2, pp. 390–421, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. E. Polak, D. Q. Mayne, and J. E. Higgins, “Superlinearly convergent algorithm for min-max problems,” Journal of Optimization Theory and Applications, vol. 69, no. 3, pp. 407–439, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Y. H. Yu and L. Gao, “Nonmonotone line search algorithm for constrained minimax problems,” Journal of Optimization Theory and Applications, vol. 115, no. 2, pp. 419–446, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J.-B. Jian, R. Quan, and X.-L. Zhang, “Generalised monotone line search algorithm for degenerate nonlinear minimax problems,” Bulletin of the Australian Mathematical Society, vol. 73, no. 1, pp. 117–127, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J.-B. Jian, R. Quan, and Q.-J. Hu, “A new superlinearly convergent SQP algorithm for nonlinear minimax problems,” Acta Mathematicae Applicatae Sinica: English Series, vol. 23, no. 3, pp. 395–410, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  15. Q.-J. Hu, Y. Chen, N.-P. Chen, and X.-Q. Li, “A modified SQP algorithm for minimax problems,” Journal of Mathematical Analysis and Applications, vol. 360, no. 1, pp. 211–222, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Z. Zhu, X. Cai, and J. Jian, “An improved SQP algorithm for solving minimax problems,” Applied Mathematics Letters, vol. 22, no. 4, pp. 464–469, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  17. F. Wang and Y. Wang, “Nonmonotone algorithm for minimax optimization problems,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6296–6308, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  18. F.-S. Wang and C.-L. Wang, “An adaptive nonmonotone trust-region method with curvilinear search for minimax problem,” Applied Mathematics and Computation, vol. 219, no. 15, pp. 8033–8041, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J.-B. Jian and M.-T. Chao, “A sequential quadratically constrained quadratic programming method for unconstrained minimax problems,” Journal of Mathematical Analysis and Applications, vol. 362, no. 1, pp. 34–45, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  20. G.-D. Ma and J.-B. Jian, “An ε-generalized gradient projection method for nonlinear minimax problems,” Nonlinear Dynamics, vol. 75, no. 4, pp. 693–700, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  21. B. Rustem, S. Žaković, and P. Parpas, “An interior point algorithm for continuous minimax: implementation and computation,” Optimization Methods & Software, vol. 23, no. 6, pp. 911–928, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  22. B. Rustem, S. Žaković, and P. Parpas, “Convergence of an interior point algorithm for continuous minimax,” Journal of Optimization Theory and Applications, vol. 136, no. 1, pp. 87–103, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  23. E. Obasanjo, G. Tzallas-Regas, and B. Rustem, “An interior-point algorithm for nonlinear minimax problems,” Journal of Optimization Theory and Applications, vol. 144, no. 2, pp. 291–318, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  24. D. Z. Du, “A gradient projection algorithm for nonlinear constraints,” Acta Mathematicae Applicatae Sinica, vol. 8, no. 1, pp. 7–16, 1985. View at Google Scholar · View at MathSciNet
  25. J. B. Jian, D. L. Han, and Q. J. Xu, “A new sequential systems of linear equations algorithm of feasible descent for inequality constrained optimization,” Acta Mathematica Sinica (English Series), vol. 26, no. 12, pp. 2399–2420, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  26. A. Vardi, “New minimax algorithm,” Journal of Optimization Theory and Applications, vol. 75, no. 3, pp. 613–634, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  27. M. J. D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations,” in Numerical Analysis, vol. 630 of Lecture Notes in Mathematics, pp. 144–157, Springer, Berlin, Germany, 1978. View at Google Scholar · View at MathSciNet