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Reflections on Geometric Programming

  • M. J. Rijckaert
  • E. J. C. Walraven
Conference paper
Part of the NATO ASI Series book series (volume 15)

Abstract

Geometric Programming is a member of the family of non-linear programming techniques, that deals with problems whose functions are all generalized polynomials. Its development started in the sixties |10|. Since then close to 500 papers have appeared on the subject, dealing with computational or theoretical aspects and with applications |26, 30|.

Keywords

Slack Variable Geometric Programming Code Hybrid Optimal Lagrange Multiplier Loose Constraint 
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.

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References

  1. [1]
    Avriel M., Dembo R., Passy U., Solution of generalized geometric programs, Int. J. of Num. Meth. in Eng., 9 (1975), 149–169CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    Avriel M., Williams A.C., Complementary geometric programming, SIM J. Appli. Math., 19 (1970), 125–141CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    Crane R.L., Hillstrom K.E., Minkoff M., Solution of the general nonlinear programming problem with subrouting VMCON, Argonne National Laboratory, Report ANL-80-64, 1980Google Scholar
  4. [4]
    Dembo R.S., Solution of complementary geometric programming problems, M. Sc. thesis, Technion, Haifa, 1972Google Scholar
  5. [5]
    Dembo R.S., Ggp — A program for solving generalized geometric programs, Users’ manual, Dept. Chem. Eng., Technion, Haifa,1972 6 DEMBO R.S., Current state of the art of algorithms and computer software for geometric programming, J.O.T.A., 26 (1978), 149–184CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Dembo R.S., Current state of the art of algorithms and computer software for geometric programming, J.O.T.A., 26 (1 978), 149–184CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Duffin R.J., Linearizing geometric programs, SIAM Review, 12 (1970), 211–227CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Duffin R.J., Peterson E.L., Geometric programs treated with slack variables, J. Appl. Anal., 2 (1972), 255–267CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Duffin R.J., Peterson E.L., Geometric programming with signomials, J.O.T.A., 11 (1973), 3–35CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Duffin R.J., Peterson E.L., Zener Cm., Geometric programming, J. Wiley, New York, London, Sydney, 1967Google Scholar
  11. [11]
    Fattler J.E., Reklaitis G.V., Sin Y.T., Root R.R., Ragsdell K.M., On the computational utility of posynomial geometric programming solution methods, Math. Prog., 22 (1982), 163–201CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Han S-P., A globally convergent method for nonlinear programming, Cornell university, Department of computer science, Report TR 75-257, 1975Google Scholar
  13. [13]
    Hock W., Schittkowski K., Test examples for nonlinear programming codes, Lecture notes in economics and mathematical systems, 187, Springer-Verlag, Berlin, Heidelberg, New York, 1980Google Scholar
  14. [14]
    Kelley J.E., The cutting plane method for solving convex programs, SIAM J. Appl. Math., 8(1960), 703–712CrossRefMathSciNetGoogle Scholar
  15. [15]
    Kochenberger G.A., Geometric programming — Extensions to deal with degrees of difficulty and loose constraints, University of Colorado, Doctoral thesis, 1969Google Scholar
  16. [16]
    Lasdon L.S., Waren A.D., Jain A., Ratner M., Design and testing of a generalized reduced gradient code for nonlinear programming, ACM Trans. on Math. Softw., 4 (1978), 34–50CrossRefzbMATHGoogle Scholar
  17. [17]
    Lasdon L.S., Waren A.D., Ratner M.W., Grg2 User’s guide, 1980Google Scholar
  18. [18]
    Martens X.M., Geometrische programmering en haar gebruik in de Chemie-ingenieurstechniek, Katholieke Universiteit Leuven, Doctoral thesis, 1976Google Scholar
  19. [19]
    Passy U., Condensing generalized polynomials, J.O.T.A., 9 (1972), 221–237CrossRefMathSciNetGoogle Scholar
  20. [20]
    Passy U., Wilde D.J., Generalized polynomial optimization, SIAM J. Appl. Math., 15 (1967), 1344–1356CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    Peterson E.L., Geometric programming, SIAM Review, 18 (1976),1–51CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    Powell M.J.D., Algorithms for nonlinear constraints that use Lagrangian functions, Math.Prog., 14 (1978), 224–248CrossRefzbMATHGoogle Scholar
  23. [23]
    Powell M.J.D., A fast algorithm for nonlinearly constrained optimization calculations, Proc. of the Dundee conference on numerical analysis, Lecture notes in mathematics, 630, Springer-Verlag, Berlin, 1878, 144–157Google Scholar
  24. [24]
    Ratner M., Lasdon L.S., Jain A., Solving geometric programs using GRG: results and comparisons, J.O.T.A., 26 (1978), 253–264CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    Rijckaert M.J., Computational Aspects of Geometric Programming, in Design and Implementation of Optimization Software, H.J. Green-berg (ed.), Sythoff & Noordhoff, Alphen a/d Rijn, 1978, 481–505Google Scholar
  26. [26]
    Rijckaert M.J., Debroey V., A bibliographical survey of geometric programming, Katholieke Universiteit Leuven, Instituut voor Chemie-ingenieurstechniek, report CE-RM-8205, 1983Google Scholar
  27. [27]
    Rijckaert M.J., Martens X.M., Numerical aspects of the use of slack variables in geometric programming, Katholieke Universiteit Leuven, Instituut voor Chemie-ingenieurstechniek, report CE-RM-7501, 1975Google Scholar
  28. [28]
    Rijckaert M.J., Martens X.M., A condensation method for generalized geometric programming, Math. Prog., 11 (1976), 89–93CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    Rijckaert M.J., Martens X.M., Gpktc — A generalized geometric programming code, Users’ manual, Katholieke Unversiteit Leuven, Instituut voor Chemie-ingenieurstechniek, report CE-RM-7601, 1976Google Scholar
  30. [30]
    Rijckaert M.J., Martens X.M., A bibliographical note on geometric programming, J.O.T.A., 26 (1978), 185–204CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    Rijckaert M.J., Martens X.M., A comparison of generalized geometric programming algorithms, J.O.T.A., 26 (1978), 205–242CrossRefzbMATHGoogle Scholar
  32. [32]
    Rijckaert M.J., Walraven E.J.C., Estimation of Lagrange multipliers in geometric programming, Op. Res., to appearGoogle Scholar
  33. [33]
    Rijckaert M.J., Walraven E.J.C., Hybrid — A generalized geometric programming code, Users’ manual, Katholieke Universiteit Leuven, Instituut voor Chemie-ingenieurstechneie, report CE-RM-8401, 1984Google Scholar
  34. [34]
    Sarma P.V.L.N., Martens X.M., Reklaitis G.V., Rijckaert M.J., A comparison of computational strategies for geometric programs, J.O.T.A., 26 (1978), 185–204CrossRefzbMATHMathSciNetGoogle Scholar
  35. [35]
    Schittkowski K., Nonlinear programming codes — Information, tests, performance, Lecture notes in economics and mathematical systems, 183, Springer-Verlag, Berlin, Heidelberg, New York, 1980Google Scholar
  36. [36]
    Walraven E.J.C., Ontwikkeling en evaluatie van primaire algoritmen voor geometrische progammering, Katholieke Universiteit Leuven, Doctoral thesis, 1984Google Scholar
  37. [37]
    Gill P.E., Murray W., The computation of Lagrange multiplier estimates for constained minimization, Math. Prog., 17 (1979) 32–60CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    Fletcher R., Practical methods of optimization — Vol. 2: Constrained optimization, J. Wiley, Chichester, New York, Brisbane, Toronto, 1981Google Scholar
  39. [39]
    Lootsma F., Performance evaluation of Nonlinear Optimization Methods via Multi-Criteria Decision Analysis and via Linear Model Analysis, in Nonlinear Optimization 1.981, M.J.D. Powell (ed), Academic Press, New York, 1982, 419–454Google Scholar
  40. [40]
    Rajasekera J.R., Peterson E.L., A posynomial geometric programming computer algorithm (PGP1), OR Report N° 194, North Carolina State University, Raleigh, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • M. J. Rijckaert
    • 1
  • E. J. C. Walraven
    • 1
  1. 1.Instituut Voor Chemie-IngenieurstechniekK.U.LeuvenLeuvenBelgium

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