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An entropy based central cutting plane algorithm for convex min-max semi-infinite programming problems

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Abstract

In this paper, we present a central cutting plane algorithm for solving convex min-max semi-infinite programming problems. Because the objective function here is non-differentiable, we apply a smoothing technique to the considered problem and develop an algorithm based on the entropy function. It is shown that the global convergence of the proposed algorithm can be obtained under weaker conditions. Some numerical results are presented to show the potential of the proposed algorithm.

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References

  1. Auslender A, Goberna M A, López M A. Penalty and smoothing methods for convex semi-infinite programming. Math Oper Res, 2009, 34: 303–319

    Article  MathSciNet  MATH  Google Scholar 

  2. Bandler J W, Liu P C, Tromp H. A nonlinear programming approach to optimal design centering, tolerancing, and tunning. IEEE Trans Circuits Syst, 1976, 23: 155–165

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertsekas D P. Constrained Optimization and Lagrange Multiplier Methods. New York: Academic Press, 1982

    MATH  Google Scholar 

  4. Chen T, Fan M K H. On convex formulation of the floor-plan area minimization problem. In: Proceedings of the International Symposium on Physical Design. Monterey, CA: Academic Press, 1998, 124–128

    Google Scholar 

  5. Elzinga J, Moore T G. A central cutting plane algorithm for the convex programming problem. Math Program, 1975, 8: 134–145

    Article  MathSciNet  MATH  Google Scholar 

  6. Fang S C, Han J Y, Huang Z H, et al. On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems. J Global Optim, 2005, 33: 369–391

    Article  MathSciNet  MATH  Google Scholar 

  7. Fang S C, Wu S Y. Solving min-max problems and linear semi-infinite programs. Comput Math Appl, 1996, 32: 87–93

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang S C, Rajasekera J R, Tsao H S J. Entropy Optimization and Mathematical Programming. Boston-London-Dordrecht: Kluwer Academic Publishers, 1997

    Book  MATH  Google Scholar 

  9. Feng Z G, Teo K L, Rehbock V. A smoothing approach for semi-infinite programming with projected Newton-type algorithm. J Ind Manag Optim, 2009, 5: 141–151

    Article  MathSciNet  MATH  Google Scholar 

  10. Fiacco A V, Kortanek K O. Semi-Infinite Programming and Applications. Lecture Notes in Economics and Mathematical Systems. Berlin: Springer, 1983

    Google Scholar 

  11. Goberna M A, López M A. Linear Semi-Infinite Optimization. New York: Wiley, 1998

    MATH  Google Scholar 

  12. Gribik P R. A central cutting plane algorithm for semi-infinite programming problems. In: Hettich R, ed. Semi-Infinite Programming, Lecture Notes in Control and Information Sciences 15. New York: Springer-Verlag, 1979, 213–232

    Chapter  Google Scholar 

  13. Hettich R, Kortanek K O. Semi-infinite programming: Theorey, methods and applications. SIAM Rev, 1993, 35: 380–429

    Article  MathSciNet  MATH  Google Scholar 

  14. Kassim A A, Vijaya Kumar B V K. Path planning autonomuous robots using neural networks. J Intell Syst, 1997, 7: 33–55

    Google Scholar 

  15. Kortanek K O, No H. A certral cutting plane algorithm for convex semi-infinite programming problems. SIAM J Optim, 1993, 3: 901–918

    Article  MathSciNet  MATH  Google Scholar 

  16. Li X. An entropy-based aggregate method for minimax optimization. Eng Optim, 1997, 18: 277–285

    Google Scholar 

  17. Li X, Fang S C. On the entropic regularization method for solving min-max problems with applications. Math Methods Oper Res, 1997, 46: 119–130

    Article  MathSciNet  MATH  Google Scholar 

  18. Polak E. On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems. Math Program, 1993, 62: 385–414

    Article  MathSciNet  MATH  Google Scholar 

  19. Polak E, Royset J O. Algorithms for finite and semi-infinite min-max-min problems using adaptive smoothing techniques. J Optim Theory Appl, 2003, 119: 421–457

    Article  MathSciNet  MATH  Google Scholar 

  20. Polak E, Royset J O. On the use of augmented Lagrangians in the solution of generalized semi-infinite min-max problems. Comput Optim Appl, 2005, 31: 173–192

    Article  MathSciNet  MATH  Google Scholar 

  21. Reemsten R, Görner S. Numerical methods for semi-infinite programming: A survey. In: Reemsten R, Rückmann J, eds. Semi-Infinite Programming. Boston, MA: Kluwer Academic Publishers, 1998, 195-27

    Google Scholar 

  22. Wu S Y, Fang S C. Solving convex programs with infinitely many linear constraints by a relaxed cutting plane method. Comput Math Appl, 1999, 38: 23–34

    Article  MathSciNet  MATH  Google Scholar 

  23. Xu S. Smoothing method for minimax problems. Comput Optim Appl, 2001, 20: 267–279

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang L P, Wu S Y, López M A. A new exchange method for convex semi-infinite programming. SIAM J Optim, 2010, 20: 2959–2977

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    LiPing Zhang & Shu-Cherng Fang

  2. Industrial Engineering and Operations Research, North Carolina State University, Raleigh, NC, 27695-7906, USA

    Shu-Cherng Fang

  3. Department of Mathematics, Cheng Kung University, Tainan, 70101, Taiwan, China

    Soon-Yi Wu

  4. Center for Theoretical Sciences, Tainan, 70101, Taiwan, China

    Soon-Yi Wu

Authors
  1. LiPing Zhang
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  2. Shu-Cherng Fang
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  3. Soon-Yi Wu
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Correspondence to LiPing Zhang or Soon-Yi Wu.

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Zhang, L., Fang, SC. & Wu, SY. An entropy based central cutting plane algorithm for convex min-max semi-infinite programming problems. Sci. China Math. 56, 201–211 (2013). https://doi.org/10.1007/s11425-012-4502-z

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  • DOI: https://doi.org/10.1007/s11425-012-4502-z

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