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Smoothing Method for Minimax Problems

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Abstract

In this paper, we propose a smoothing method for minimax problem. The method is based on the exponential penalty function of Kort and Bertsekas for constrained optimization. Under suitable condition, the method is globally convergent. Preliminary numerical experiments indicate the promising of the algorithm.

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References

  1. J. Asaadi, “A computational comparison of some non-linear programs,” Math. Programming, 1973.

  2. J.W. Bandler and C. Charalambous, “Nonlinear programming using minimax techniques,” J. Optimization Theory Appl., vol. 13, pp. 607–619, 1974.

    Google Scholar 

  3. D.P. Bertsekas, “Minimax methods based on approximation,” in Proceeding of the 1976 Johns Hopkins Conference on Information Sciences and Systems, 1976.

  4. D.P. Bertsekas, “A new algorithm for solution of nonlinear resistive networks involving diodes,” IEEE Transactions on Circuit Theory, vol. CAS 23, pp. 599–608, 1976.

    Google Scholar 

  5. D.P. Bertsekas, “Approximation Procedures based on the method of multipliers,” J. Optimization Theory and Applications, vol. 23, pp. 487–510, 1977.

    Google Scholar 

  6. J. Burke and S. Xu, “A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem,” Mathematical Programming, vol. 87, pp. 113–130, 2000.

    Google Scholar 

  7. J. Burke and S. Xu, “The global linear convergence of a non-interior path-following algorithm for linear complementarity problem,” Mathematics of Operations Research, vol. 23, pp. 719–734, 1998.

    Google Scholar 

  8. C. Charalamous and J.W. Bandler, “Nonlinear minimax optimization as a sequence of least pth optimization with finite values of p,” Internat. J. Systems Sci., vol. 7, pp. 377–391, 1976.

    Google Scholar 

  9. C. Charalamous and A.R. Conn, “An efficient method to solve the minimax problem directly,” SIAM J. Numer. Anal, vol. 15, pp. 162–187, 1978.

    Google Scholar 

  10. B. Chen and N. Xiu, “A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions,” SIAM J. Optimization, Vol. 9, pp. 605–623, 1999.

    Google Scholar 

  11. V.F. Demyanov and V.N. Molozemov, Introduction to Minimax, Wiley: New York, 1974.

    Google Scholar 

  12. D.Z. Du and P.M. Pardalos (eds), Minimax and Applications, Kluwer Academic Publishers: Dordrecht, 1995.

    Google Scholar 

  13. F. Facchinei, H. Jiang and L. Qi, “A smoothing method for mathematical programs with equilibrium constraints,” Mathematical Programming, vol. 85, pp. 107–134, 1999.

    Google Scholar 

  14. M. Fukushima, Z.-Q. Luo and J.-S. Pang, “Aglobally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,” Comput. Optim. Appl., vol. 10, pp. 5–34, 1998.

    Google Scholar 

  15. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

  16. K. Hotta and A. Yoshise, “Global convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for nonlinear complementarity problems,” Mathematical Programming, Vol. 86, pp. 105–133, 1999.

    Google Scholar 

  17. B.W. Kort and D.P. Bertsakas, “A new penalty function algorithm algorithm for constrained minimization,” in Proceedings of the 1972 IEEE conferences on Decision and Control, New Orlean, Louisiana, 1972.

  18. X.S. Li, “An aggregate function method for nonlinear programming,” Science in China (A), vol. 34, pp. 1467–1473, 1991.

    Google Scholar 

  19. Z.-H. Lin and Hou-Duo Qi, “A non-interior homotopy system for generalized nonlinear complementarity problem,” Preprint of the State key laboratory of scientific and engineering computing, Academic Sinica, Beijing, China, 1998.

    Google Scholar 

  20. W. Murray and M.L. Overton, “Aprojected Lagrangian algorithm for nonlinear minimax optimization,” SIAM Journal on Scientific and Statistic Computing, vol. 1, pp. 345–370, 1980.

    Google Scholar 

  21. M.R. Osborne and G.A. Watson, “An algorithm for minimax approximation in the non-linear case,” Comput. J., vol. 12, pp. 63–68, 1969.

    Google Scholar 

  22. J.-M. Peng and Z. Lin, “Anon-interior continuation method for generalized linear complementarity problems,” Mathematical Programming, Vol. 86, pp. 533–563, 1999.

    Google Scholar 

  23. E. Polak, “On the mathematical foundations of nondifferentiable optimization,” SIAM Review, vol. 29, pp. 21–89, 1987.

    Google Scholar 

  24. E. Polak, Optimization: Algorithm and Consistent Approximations, Springer Verlag: New York, 1997.

    Google Scholar 

  25. E. Polak and J.E. Higgins and D.Q. Mayne, “A barrier function method for minimax problems,” Mathematical Programming, vol. 64, pp. 277–294, 1994.

    Google Scholar 

  26. L. Qi and D. Sun and G. Zhou, “A new look at smoothing Newton methods for non-linear complementarity problems and box constrained variational inequalities,” Mathematical Programming, vol. 87, pp. 1–35, 2000.

    Google Scholar 

  27. R.T. Rockafellar, “Linear-quadric programming and optimal control,” SIAM Journal on Control and Optimization, vol. 25, pp. 781–814, 1987.

    Google Scholar 

  28. R.T. Rockafellar, “Computational schemes for large-scale problems in extended linear-quadratic programming,” Mathematical Programming, vol. 48, pp. 447–474, 1990.

    Google Scholar 

  29. S.E. Sussman-Fort, “Approximate direct-search minimax circuit optimization,” International Journal for Numerical Methods in Engineering, vol. 28, pp. 359–368, 1989.

    Google Scholar 

  30. P. Tseng, “Analysis of a non-interior continuation method based on chen-mangasarian smoothing functions for complementarity problems,” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Masao Fukushima and Liqun Qi, (Eds.), Kluwer Academic. Publishers: Dordrecht, 1999, pp. 381–404.

    Google Scholar 

  31. A.D. Warren, L.S. Lasdon and D.F. Suchman, “Optimization in engineering design,” Proc. IEEE, vol. 55, pp. 1885–1897, 1967.

    Google Scholar 

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Xu, S. Smoothing Method for Minimax Problems. Computational Optimization and Applications 20, 267–279 (2001). https://doi.org/10.1023/A:1011211101714

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  • DOI: https://doi.org/10.1023/A:1011211101714