Optimization under Uncertainty and Linear Semi-Infinite Programming: A Survey

  • Teresa León
  • Enriqueta Vercher
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 57)

Abstract

This paper deals with the relationship between semi-infinite linear programming and decision making under uncertainty in imprecise environments. Actually, we have reviewed several set-inclusive constrained models and some fuzzy programming problems in order to see if they can be solved by means of a linear semi-infinite program. Finally, we present some numerical examples obtained by using a primal semi-infinite programming method.

Keywords

Membership Function Fuzzy Number Extreme Point Linear Programming Problem Triangular Fuzzy Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E.J. Anderson. A new primal algorithm for semi-infinite linear programming. In E. J. Anderson and A. B. Philpott, editors, Infinite Programmtmning, pages 108–122. Springer Verlag, 1985.CrossRefGoogle Scholar
  2. [2]
    E. J. Anderson and A. Lewis. An extension of the simplex algorithm for semi-infinite linear programming, Mathematical Programming, A44: 247–269, 1989 .MathSciNetCrossRefGoogle Scholar
  3. [3]
    C. H. Carlsson and P. Korhonen. A parametric approach to fuzzy linear programming, Fuzzy Sets and Systems, 20: 17–30, 1986.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    G. B. Dantzig. Linear Programming and Extensions, Princeton University Press, 1963.MATHGoogle Scholar
  5. [5]
    D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications, Academic Press, 1980.MATHGoogle Scholar
  6. [6]
    J. E. Falk. Exact solutions of inexact linear programs, Operations Research, 24: 783–787, 1976.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    S. C. Fang, C. F Hu, H. F. Wang and S. Y. Wu. Linear Programming with fuzzy coefficients in constraints, Computers and Mathematics with Applications, 37: 63–76, 1999.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    K. Glashoff and S. A. Gustafson. Linear Optimization and Approximation, Springer-Verlag, 1983.CrossRefMATHGoogle Scholar
  9. [9]
    M. A. Goberna and V. Jornet. Geometric fundamentals of the simplex method in semi-infinite programming, OR Spektrum, 10: 145–152, 1988.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. A. Goberna and M. A. Lopez. Linear Semi-Infinite Optimization, Wiley, 1998.MATHGoogle Scholar
  11. [11]
    R. Hettich and K. O. Kortanek. Semi-infinite programming: theory, methods and applications, SIAM Review, 35: 380–429, 1993.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    C. F. Hu and S. C. Fang. A relaxed cutting plane algorithm for solving fuzzy inequality systems, Optimization, 45: 86–106, 1999.MathSciNetGoogle Scholar
  13. [13]
    P. Kall and S. W. Wallace. Stochastic Programming, Wiley, 1994.MATHGoogle Scholar
  14. [14]
    Y. J. Lai and Ch. L. Hwang. Fuzzy Mathemmatical Programmming. Methods and Applications, Springer-Verlag, 1992.CrossRefGoogle Scholar
  15. [15]
    T. León and E. Vercher. An optimality test for semi-infinite linear programming, Optimization, 26: 51–60, 1992.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    T. León and E. Vercher. New descent rules for solving the linear semiinfinite programming problem, Operations Research Letters, 15: 105–114, 1994.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    T. León, S. Sanmatías and E. Vercher. Un método primal de optimización semi-infinita para la aproximación uniforme de funciones, Qüestiió, 22: 313–335, 1998.MATHGoogle Scholar
  18. [18]
    T. León, S. Sanmatías and E. Vercher. On the numerical treatment of linearly constrained semi-infinite optimization problems, European Journal of Operational Research, 121: 78–91, 2000.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    P. Nash. Algebraic fundamentals of linear programming. In E. J. Anderson and A. B. Philpott, editors, Infinite Programming, pages 37–52, SpringerVerlag, 1985.CrossRefGoogle Scholar
  20. [20]
    C. V. Negoita, P. Flondor and M. Sularia. On fuzzy environment in optimization problems, Econommtic Computer and Econommic Cybernetic Studies and Researches, 2: 13–24, 1977.MathSciNetGoogle Scholar
  21. [21]
    M. L. Parks and A. L. Soyster. Semi-infinite and fuzzy set programming. In A. V. Fiacco and K.O. Kortanek, editors, Semi-Infinite Programming and Applications, pages 219–235, Springer-Verlag, 1983.CrossRefGoogle Scholar
  22. [22]
    E. Polak and L. He. Unified steerable phase I-phase II of feasible directions for semi-infinite optimization, Journal of Optimization Theory and Applications, 69: 83–107, 1991.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    J. Ramik and J. Rimanek. Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems, 16: 123–138, 1985.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    R. Reemtsen and S. Görner. Numerical methods for semi-infinite programming: A survey. In Semi-Infinite Programming, pages 195–275, Kluwer Academic, 1998.CrossRefGoogle Scholar
  25. [25]
    R. Reemtsen and J. J Rückmann (editors). Semi-Infinite Programming, Kluwer, 1998.MATHGoogle Scholar
  26. [26]
    R. Roleff. A stable multiple exchange algorithm for linear SIP. In R. Hettich, editor, Semi-Infinite Programming, pages 83–96, Springer-Verlag, 1979.CrossRefGoogle Scholar
  27. [27]
    A. L. Soyster. Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 21: 1154–1157, 1973.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    A. L. Soyster. A Duality theory for convex programming with set-inclusive constraints, Operations Research, 22: 892–898, 1974.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    A. L. Soyster. Inexact linear programming with generalized resource sets, European Journal of Operational Research, 3: 316–321, 1979.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    H. Tanaka, H. Ichihashi and K. Asai. A formulation of fuzzy linear programming problems based on comparison of fuzzy numbers, Control and Cybernetics, 13: 186–194, 1984.MathSciNetGoogle Scholar
  31. [31]
    D. J. Thuente. Duality theory for generalized linear programs with computational methods, Operations Research, 28: 1005–1011, 1980.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    R. Tichatschke, R. Hettich and G. Still. Connections between generalized, inexact and semi-infinite linear programming, ZOR-Methods and Models of Operations Research, 33: 367–382, 1989.MathSciNetMATHGoogle Scholar
  33. [33]
    R. J. B. Wets and W. T. Ziemba (editors). Stochastic Programming. State of the Art, 1998, Annals of Operations Research 85, 1999.Google Scholar
  34. [34]
    L. A. Zadeh. Fuzzy sets, Information and Control, 8: 338–353, 1965.MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    H. J. Zimmermann. Fuzzy set theory and its applications (3rd ed. ), Kluwer, 1996.CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Teresa León
    • 1
  • Enriqueta Vercher
    • 1
  1. 1.Departament d’Estadística i Investigació OperativaUniversitat de ValènciaValenciaSpain

Personalised recommendations