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Semismooth Newton Methods for Solving Semi-Infinite Programming Problems

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Abstract

In this paper we present some semismooth Newton methods for solving the semi-infinite programming problem. We first reformulate the equations and nonlinear complementarity conditions derived from the problem into a system of semismooth equations by using NCP functions. Under some conditions a solution of the system of semismooth equations is a solution of the problem. Then some semismooth Newton methods are proposed for solving this system of semismooth equations. These methods are globally and superlinearly convergent. Numerical results are also given.

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References

  1. Chen, X., Qi, L. and Sun, D. 1998, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comput. 67, 519-540.

    Google Scholar 

  2. Clarke, F.H. 1983, Optimization and Nonsmooth Analysis, Wiley, New York.

    Google Scholar 

  3. Conn, A.R. and Gould, N.I.M. 1987, An exact penalty function for semi-infinite programming, Math. Programming 37, 19-40.

    Google Scholar 

  4. Coope, I.D. and Watson, G.A. 1985, A projected Lagrangian algorithm for semi-infinite programming, Math. Programming 32, 337-356.

    Google Scholar 

  5. De Luca, T., Facchinei, F. and Kanzow, C. 1996, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programming 75, 407-439.

    Google Scholar 

  6. Facchinei, F. and Kanzow, C. 1997, A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems, Math. Programming 76, 493-512.

    Google Scholar 

  7. Fischer, A., 1997, Solution of monotone complementarity problems with locally Lipschitzian functions, Math. Programming 76, 513-532.

    Google Scholar 

  8. Janin, R., 1984, Direction derivative of the marginal function in nonlinear programming, Math. Programming Study 21, 127-138.

    Google Scholar 

  9. Jiang, H., 1999, Global convergence analysis of the generalized Newton and Gauss-Newton methods of the Fischer-Burmeister equation for the complementarity problem, Math. Oper. Res. 24, 529-543.

    Google Scholar 

  10. Jiang, H. and Qi, L. 1997, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM J. Control and Optimization 35, 178-193.

    Google Scholar 

  11. Li, D., Qi, L., Tam, J. and Wu, S.Y. Smoothing Newton methods for solving semi-infinite programming problems, to appear in: J. Global Optimization.

  12. Mangasarian, O.L., 1969 Nonlinear Programming, McGraw-Hill, New York, 1969.

    Google Scholar 

  13. Mayne, D.Q., Michalska, H. and Polak, E. 1994, An efficient algorithm for solving semi-infinite inequality problems with box constraints, J. Appl. Math. and Optim. 30, 135-157.

    Google Scholar 

  14. Mayne, D.Q. and Polak, E. 1982, A quadratically convergent algorithm for solving infinite dimensional inequalities, J. Appl. Math. and Optim. 9, 25-40.

    Google Scholar 

  15. Pang, J.-S. and Qi, L. 1993, Nonsmooth equations: Motivation and algorithms, SIAM J. Optimization 3, 443-465.

    Google Scholar 

  16. Polak, E., 1997, Optimization: Algorithms and Consistent Approximation, Springer-Verlag, New York.

    Google Scholar 

  17. Polak, E., and Tits, A. 1982, A recursive quadratic programming algorithm for semi-infinite optimization problems, Appl. Math. Optim. 8, 325-349.

    Google Scholar 

  18. Qi, L. 1993, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res. 18 227-244.

    Google Scholar 

  19. Qi, L., 1999, Regular pseudo-smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems, Math. Oper. Res. 24, 440-471.

    Google Scholar 

  20. Qi, L. and Jiang, H. 1997, Semismooth Karush-Kuhn-Tucker equations and convergence analysis of Newton methods and quasi-Newton methods for solving these equations, Math. Oper. Res. 22, 301-325.

    Google Scholar 

  21. Qi, L., Sun, D. and Zhou, G. 2000, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Programming 87, 1-35.

    Google Scholar 

  22. Qi, L. and Sun, J. 1993, A nonsmooth version of Newton’s method, Math. Programming 58, 353-367.

    Google Scholar 

  23. Qi, L., and Wei, Z. 2000, On the constant positive linear dependence condition and its applications to SQP methods, SIAM J. Optimization 10, 963-981.

    Google Scholar 

  24. Reemsten, R. and Rückmann, J. 1998, Semi-Infinite Programming, Kluwer Academic Publishers, Boston.

    Google Scholar 

  25. Rückmann, J.J. and Shapiro, A. 2001, Second Order Optimality Conditions in Generalized Semi-Infinite Programming, Set-Valued Analysis 9, 169-186.

    Google Scholar 

  26. Shapiro, A. 1998, First and second order optimality conditions and perturbation analysis of semi-infinite programming problems, in: R. Reemsten and J. Rückmann, eds., pp. 103–133 Semi-Infinite Programming, Kluwer Academic Publishers, Boston.

    Google Scholar 

  27. Tanaka, Y., Fukushima, M. and Ibaraki, T. 1988, A globally convergent SQP method for semiinfinite nonlinear optimization, J. Comput. Appl. Math. 23, 141-153.

    Google Scholar 

  28. Tanaka, Y., Fukushima, M. and Ibaraki, T. 1988, A comparative study of several semi-infinite nonlinear programming algorithms, European J. Oper. Res. 36, 92-100.

    Google Scholar 

  29. Teo, K.L., Yang, X.Q. and Jennings, L.S. 2001, Computational discretization algorithms for functional inequality constrained optimization, Ann. Oper. Res. 98, 215-234.

    Google Scholar 

  30. Watson, G.A., 1981, Globally convergent methods for semi-infinite programming, BIT, 21 362-373.

    Google Scholar 

  31. Watson, G.A., 1983, Numerical experiments with globally convergent methods for semi-infinite programming problems, in: Semi-infinite programming and applications, A.V. Fiacco and K.O. Kortanek, eds., (Austin, Tex., 1981), 193-205, Lecture Notes in Econom. and Math. Systems, 215, Springer, Berlin-New York.

    Google Scholar 

  32. Yamashita, N. and Fukushima, M. 1997, Modified Newton methods for solving semismooth reformulations of monotone complementarity problems, Math. Programming 76, 469-491.

    Google Scholar 

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Qi, L., Wu, SY. & Zhou, G. Semismooth Newton Methods for Solving Semi-Infinite Programming Problems. Journal of Global Optimization 27, 215–232 (2003). https://doi.org/10.1023/A:1024814401713

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