Semi-Infinite Programming: Recent Trends of Development

  • Sven-Åke Gustafson
  • Kenneth O. Kortanek

Abstract

It is well accepted that linear programming (LP) serves as an ideal for both theory and computation within the broad field of optimization. Semi-infinite programming is a next level of extension of LP that allows finitely many variables to appear in infinitely many constraints. During the last ten years, a series of international meetings has reconfirmed that the theoretical and practical manifestations and applications of this problem formulation are abundant and significant.

Keywords

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References

  1. [1]
    Anderson, E.J. and P. Nash, Linear Programming in Infinite-dimensional Spaces, Wiley, Chichester, 1987.Google Scholar
  2. [2]
    Anderson, E.J. and A.B. Philpott (eds.), Infinite Programming, Springer-Verlag, Berlin, 1985.Google Scholar
  3. [3]
    Andersson, R. et al., “The Automatic Generation of Convex Surfaces,” Volvo Data AB, Gothenburg, Sweden, 1987.Google Scholar
  4. [4]
    Ben Israel, A., A. Chames, and K.O. Kortanek, “Duality and Asymptotic Solvability Over Cones,” Bulletin of the American Mathematical Society 75 (1969), 318–324.CrossRefGoogle Scholar
  5. [5]
    Ben-Israel, A., A. Chames, and K.O. Kortanek, “Erratum to ”Duality and Asymptotic Solvability Over Cones“, Bulletin of the American Mathematical Society 76 (1970), 426.CrossRefGoogle Scholar
  6. [6]
    Ben-Israel, A., A. Chames, and K.O. Kortanek, “Asymptotic Duality in Semi-Infinite Programming and the Convex Core Topology,” Ren. di Mat. 4 (1971), 1–17, Oderisi Gubbio.Google Scholar
  7. [7]
    Ben-Israel, A., A. Chames, and K.O. Kortanek, “Asymptotic Duality Over Closed Convex Sets,” Journal of Mathematical Analysis and Applications 35 (1971), 677–691.CrossRefGoogle Scholar
  8. [8]
    Chames, A., W.W. Cooper, and K.O. Kortanek, “A Duality Theory for Convex Programs with Convex Constraints,” Bulletin of the American Mathematical Society 68 (1962), 605–608.CrossRefGoogle Scholar
  9. [9]
    Chames, A., W.W. Cooper, and K.O. Kortanek, “Duality, Haar Programs and Finite Sequence Spaces”, Proceedings of the National Academy of Sciences U.S. 48 (1962), 783–786.CrossRefGoogle Scholar
  10. [10]
    Chames, A., W.W. Cooper, and K.O. Kortanek, “Duality in Semi-Infinite Programs and Some Works of Haar and Caratheodory,” Management Science 9 (1963), 209–228.CrossRefGoogle Scholar
  11. [11]
    Chames, A., W.W. Cooper, and K.O. Kortanek, “On Representation of Semi-Infinite Programs Which Have No Duality Gaps,” Management Science 12 (1965), 113–121.CrossRefGoogle Scholar
  12. [12]
    Chames, A., W.W. Cooper, and K.O. Kortanek, “On the Theory of Semi-Infinite Programming and a Generalization of the Kuhn—Tucker Saddle Point Theorem for Arbitrary Convex Functions,” Naval Research Logistics Quarterly 16 (1969), 41–51.Google Scholar
  13. [13]
    Chames, A., P.R. Gribik, and K.O. Kortanek, “Separably Infinite Programs,” Z. Operations Research 24 (1980), 33–45.Google Scholar
  14. [14]
    Chames, A., P.R. Gribik, and K.O. Kortanek, “Polyextremal Principles and Separably-Infinite Programs,” Z. Operations Research 25 (1980), 211–234.Google Scholar
  15. [15]
    Chames, A., K.O. Kortanek, and V. Lovegren, “A Saddle Value Characterization of Fan’s Equilibrium Points,” in A.V. Fiacco, and K.O. Kortanek, (eds), Semi-Infinite Programming and Applications, Springer-Verlag, Berlin, 1983, 37–49.Google Scholar
  16. [16]
    Chames, A., K.O. Kortanek, and W. Raike, “Extreme Point Solutions in Mathematical Programming: An Opposite Sign Algorithm,” SRM No. 129, Northwestern University, June 1965.Google Scholar
  17. [17]
    Chames, A., K.O. Kortanek, and S. Thore, “An Infinite Constrained Game Duality Characterizing Economic Equilibrium,” Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA, March 1981.Google Scholar
  18. [18]
    Cheney, E.W., Introduction to Approximation Theory, McGraw-Hill, New York, 1966.Google Scholar
  19. [19]
    Conn, A.R. and N.I.M. Gould, “An Exact Penalty Function for Semi-Infinite Programming,” Mathematical Programming 37 (1987), 19–40.CrossRefGoogle Scholar
  20. [20]
    Coope, I.D., G.A. Watson, “A Projected Lagrangian Algorithm for Semi-Infinite Programming,” Mathematical Programming 32 (1985), 337–356.CrossRefGoogle Scholar
  21. [21]
    Duffin, R.J., L.A. Karlowitz, “An Infinite Linear Program with a Duality Gap,” Management Science 12 (1965), 122–134.CrossRefGoogle Scholar
  22. [22]
    Fahlander, K., “Computer Programs for Semi-Infinite Optimization,” TRITANA-7312, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-10044 Stockholm 70, Sweden, 1973.Google Scholar
  23. [23]
    Ferris, M.C. and A.B. Philpott, “An Interior Point Algorithm for Semi-Infinite Linear Programming,” Mathematical Programming 43 (1989) 257–276.CrossRefGoogle Scholar
  24. [24]
    Fiacco, A.V. and K.O. Kortanek (eds.), Semi-Infinite Programming and Applications, Springer-Verlag, Berlin, 1983.Google Scholar
  25. [25]
    Georg, K. and R. Hettich, “On the Numerical Stability of the Simplex Algorithm,” Optimization (Ilmenau) 18 (1987) 361–372.CrossRefGoogle Scholar
  26. [26]
    Glashoff, K. and S.-A. Gustafson, Linear Optimization and Approximation, Springer-Verlag, New York, 1983.CrossRefGoogle Scholar
  27. [27]
    Gustafson, S.-A., “On the Computational Solution of a Class of Generalized Moment Problems,” SIAM Journal of Numerical Analysis 7 (1970), 343–357.CrossRefGoogle Scholar
  28. [28]
    Gustafson, S.-A., “Investigating Semi-Infinite Programs Using Penalty Functions and Lagrangian Methods,” Journal of the Australian Mathematical Society,Series B 28 (1986), 158–169.CrossRefGoogle Scholar
  29. [29]
    Gustafson, S.-A. and K.O. Kortanek, “Numerical Treatment of a Class of Semi-Infinite Programming Problems,” Naval Research Logistics Quarterly 20 (1973), 477–504.CrossRefGoogle Scholar
  30. [30]
    Gustafson, S.-A and K.O. Kortanek, “Semi-Infinite Programming and Applications,” in A. Bachem, M. Grotschel, and B. Korte (eds.), Mathematical Programming—The State of the Art,“ Bonn 1982, Springer-Verlag, Berlin 1983.Google Scholar
  31. [31]
    Hettich, R. (ed.), “Semi-Infinite Programming,” Lecture Notes in Control and Information Sciences 15, Springer-Verlag, Berlin, 1979.Google Scholar
  32. [32]
    Hettich, R., “An Implementation of a Discretization Method for Semi-Infinite Programming,” Mathematical Programming 34 (1986), 354–361.CrossRefGoogle Scholar
  33. [33]
    Hettich, R., “On the Computation of Membrane-Eigenvalues by Semi-Infinite Programming Methods,” in E.J. Anderson and A.B. Philpott (eds.), Infinite Programming, Lecture Notes in Economics and Math. Systems 259, Springer-Verlag, Berlin-Herdelberg-New York, 1985,79–89.CrossRefGoogle Scholar
  34. [34]
    Hettich, R. and P. Zencke, Numerische Methoden der Approximation und Semiinfiniten Optimierung, Teubner, Stuttgart, 1982.Google Scholar
  35. [35]
    Karmarkar, N., “A New Polynomial Time Algorithm for Linear Programming,” Combinatorica 4 (1984), 373–375.CrossRefGoogle Scholar
  36. [36]
    Karlin, S. and W.J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience Publishers, New York, 1966.Google Scholar
  37. [37]
    Kortanek, K.O., “A Biography of Professor A. Chames and His Scientific Work,” in Extremal Methods and Systems Analysis-An International Symposium on the Occasion of Professor Abraham Chames’ Sixtieth Birthday, A.V. Fiacco and K.O. Kortanek (eds.), Lecture Notes in Economics and Mathematical Systems 174(1980), 2–9.Google Scholar
  38. [38]
    Kortanek, K.O., “Vector-Supercomputer Experiments With the Linear Programming Scaling Algorithm,” forthcoming, SIAM Journal of Scientific and Statistical Computing. Google Scholar
  39. [39]
    Kortanek, K.O. and Z. Jishan, “New Purification Algorithms for Linear Programming,” Naval Research Logistics Quarterly (in press).Google Scholar
  40. [40]
    ’Crabs, W., Optimierung und Approximation,Teubner, Stuttgart, 1975.Google Scholar
  41. [41]
    Pietrzykowski, T., “An Exact Potential Function for Constrained Maxima,” SIAM Journal of Numerical Analysis 6 (1969), 299–304.CrossRefGoogle Scholar
  42. [42]
    Polak, E., “On the Mathematical Foundation of Nondifferentiable Optimization in Engineering Design,” SIAM Review 29 (1987), 21–90.CrossRefGoogle Scholar
  43. [43]
    Remez, E. Ya, “Sur le Calcul Effectif des Polynomes d’Approximation de Tchebichef,” Comptes Rendus Hebdomadaries, l’Academie des Sciences, Paris, 199 (1934), 337–340.Google Scholar
  44. [44]
    Zencke, P. and R. Hettich, “Directional Derivatives for the Value-Function in Semi-Infinite Programming,” Mathematical Programming 38 (1987), 323–340.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Sven-Åke Gustafson
  • Kenneth O. Kortanek

There are no affiliations available

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