Advertisement

Branch-and-Bound Reduction Type Method for Semi-Infinite Programming

  • Ana I. Pereira
  • Edite M. G. P. Fernandes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6784)

Abstract

Semi-infinite programming (SIP) problems can be efficiently solved by reduction type methods. Here, we present a new reduction method for SIP, where the multi-local optimization is carried out with a multi-local branch-and-bound method, the reduced (finite) problem is approximately solved by an interior point method, and the global convergence is promoted through a two-dimensional filter line search. Numerical experiments with a set of well-known problems are shown.

Keywords

Nonlinear Optimization Semi-Infinite Programming Global Optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    El-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the Newton interior-point method for nonlinear programming. Journal of Optimization Theory and Applications 89, 507–541 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ben-Tal, A., Teboule, M., Zowe, J.: Second order necessary optimality conditions for semi-infinite programming problems. Lecture Notes in Control and Information Sciences, vol. 15, pp. 17–30 (1979)Google Scholar
  3. 3.
    Coope, I.D., Watson, G.: A projected Lagrangian algorithm for semi-infinite programming. Mathematical Programming 32, 337–356 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Mathematical Programming 91, 239–269 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Goberna, M.A., López, M.A. (eds.): Semi-Infinite Programming. Recent Advances in Nonconvex Optimization and Its Applications. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  6. 6.
    Hendrix, E.M.T., G-Tóth, B.: Introduction to nonlinear and global optimization. Springer optimization and its applications, vol. 37. Springer, Heidelberg (2010)zbMATHGoogle Scholar
  7. 7.
    Hettich, R., Jongen, H.T.: Semi-infinite programming: conditions of optimality and applications. In: Stoer, J. (ed.) Lectures Notes in Control and Information Science - Optimization Techniques, vol. 7, pp. 1–11. Springer, Heidelberg (1978)Google Scholar
  8. 8.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming: Theory, methods and applications. SIAM Review 35, 380–429 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Horst, R., Tuy, H.: Global optimization. deterministic approaches. Springer, Heidelberg (1996)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ingber, L.: Very fast simulated re-annealing. Mathematical and Computer Modelling 12, 967–973 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Li, D.-H., Qi, L., Tam, J., Wu, S.-Y.: A smoothing Newton method for semi-infinite programming. Journal of Global Optimization 30, 169–194 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ling, C., Ni, Q., Qi, L., Wu, S.-Y.: A new smoothing Newton-type algorithm for semi-infinite programming. Journal of Global Optimization 47, 133–159 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Liu, G.-x.: A homotopy interior point method for semi-infinite programming problems. Journal of Global Optimization 37, 631–646 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    López, M., Still, G.: Semi-infinite programming. European Journal of Operations Research 180, 491–518 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Parsopoulos, K., Plagianakos, V., Magoulas, G., Vrahatis, M.: Objective function stretching to alleviate convergence to local minima. Nonlinear Analysis 47, 3419–3424 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Pereira, A.I.P.N., Fernandes, E.M.G.P.: On a reduction line search filter method for nonlinear semi-infinite programming problems. In: Sakalauskas, L., Weber, G.W., Zavadskas, E.K. (eds.) Euro Mini Conference Continuous Optimization and Knowledge-Based Technologies, vol. 9, pp. 174–179 (2008), ISBN: 978-9955-28-283-9Google Scholar
  18. 18.
    Pereira, A.I.P.N., Fernandes, E.M.G.P.: An Hyperbolic Penalty Filter Method for Semi-Infinite Programming. Numerical Analysis and Applied Mathematics. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) AIP Conference Proceedings, vol. 1048, pp. 269–273. Springer, Heidelberg (2008)Google Scholar
  19. 19.
    Pereira, A.I.P.N., Fernandes, E.M.G.P.: A reduction method for semi-infinite programming by means of a global stochastic approach. Optimization 58, 713–726 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Price, C.J., Coope, I.D.: Numerical experiments in semi-infinite programming. Computational Optimization and Applications 6, 169–189 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Qi, L., Wu, W.S.-Y., Zhou, G.: Semismooth Newton methods for solving semi-infinite programming problems. Journal of Global Optimization 27, 215–232 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Qi, L., Ling, C., Tong, X., Zhou, G.: A smoothing projected Newton-type algorithm for semi-infinite programming. Computational Optimization and Applications 42, 1–30 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Reemtsen, R., Rückmann, J.-J.: Semi-infinite programming. Nonconvex Optimization and Its Applications, vol. 25. Kluwer Academic Publishers, Dordrecht (1998)zbMATHGoogle Scholar
  24. 24.
    Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods, Mathematical Programming Ser. B 87, 303–316 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Silva, R., Ulbrich, M., Ulbrich, S., Vicente, L.N.: A globally convergent primal-dual interior-point filter method for nonlinear programming: new filter optimality measures and computational results, preprint 08-49, Dept. Mathematics, U. Coimbra (2008)Google Scholar
  26. 26.
    Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM Journal on Control and Optimization 42, 769–788 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Tanaka, Y., Fukushima, M., Ibaraki, T.: A comparative study of several semi-infinite nonlinear programmnig algorithms. European Journal of Operations Research 36, 92–100 (1988)CrossRefzbMATHGoogle Scholar
  28. 28.
    Ulbrich, M., Ulbrich, S., Vicente, L.N.: A globally convergent primal-dual interior-point filter method for nonlinear programming. Mathematical Programming 100, 379–410 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications 13, 231–252 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Vaz, A.I.F., Fernandes, E.M.G.P., Gomes, M.P.S.F.: Robot trajectory planning with semi-infinite programming. European Journal of Operational Research 153, 607–617 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Vaz, A.I.F., Ferreira, E.C.: Air pollution control with semi-infinite programming. Applied Mathematical Modelling 33, 1957–1969 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Vázquez, F.G., Rückmann, J.-J., Stein, O., Still, G.: Generalized semi-infinite programming: a tutorial. Journal of Computational and Applied Mathematics 217, 394–419 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM Journal on Optimization 16, 1–31 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: local convergence. SIAM Journal on Optimization 16, 32–48 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming 106, 25–57 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Weber, G.-W., Tezel, A.: On generalized semi-infinite optimization of genetic network. TOP 15, 65–77 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Yi-gui, O.: A filter trust region method for solving semi-infinite programming problems. Journal of Applied Mathematics and Computing 29, 311–324 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ana I. Pereira
    • 1
  • Edite M. G. P. Fernandes
    • 2
  1. 1.Polytechnic Institute of BragançaBragançaPortugal
  2. 2.Algoritmi R&D CentreUniversity of MinhoBragaPortugal

Personalised recommendations