Skip to main content
SpringerLink
Log in

Geometric programming

Method and applications

  • Survey
  • Published:
Operations-Research-Spektrum Aims and scope Submit manuscript

Summary

This paper is a survey of the theory and applications of geometric programming. After an introduction to the theory of geometric programming, the economic interpretation of duality, transformation of certain optimization problems into standard geometric programming problems and some extensions of posynomial geometric programming will be described. Finally, an overview of the algorithms and applications of geometric programming with focus on economics and management science will be given.

Zusammenfassung

Dieser Beitrag gibt einen Überblick über die Theorie und Anwendungen der geometrischen Programmierung. Nach einer Einführung in die Theorie der geometrischen Programmierung werden die ökonomische Interpretation der Dualität, die Transformation bestimmter Optimierungsprobleme in ein Standardproblem der geometrischen Programmierung sowie einige Weiterentwicklungen der posinomischen geometrischen Programmierung beschrieben. Die abschließenden Abschnitte geben einen Überblick über die Algorithmen und Anwendungen der geometrischen Programmierung mit Schwerpunkt in Ökonomie und Management Science.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abrams R (1975) Optimization models for regional air pollution control. In: Charnes A, Synn WR (eds) Mathematical analysis of decision problems in ecology. Springer, Berlin Heidelberg New York

    Google Scholar 

  2. Avriel M, Rijckaert MJ, Wilde DJ (eds) (1973) Optimization and design. Prentice-Hall, Englewood Cliffs, NJ

    Google Scholar 

  3. Avriel M, Wilde DJ (1970) Stochastic geometric programming. In: Kuhn HW (ed) Proceedings of the Princeton symposium on mathematical programming. Princeton University Press, Princeton, NJ, pp 73–91

    Google Scholar 

  4. Balachandran V, Gensch D (1974) Solving the marketingmix problem using geometric programming. Manag Sci 21: 160–171

    Article  Google Scholar 

  5. Beck PA, Ecker JG (1975) A modified concave simplex algorithm for geometric programming. JOTA 15:189–202

    Article  Google Scholar 

  6. Beightler CS, Phillips DT (1976) Applied geometric programming. John Wiley & Sons, New York London

    Google Scholar 

  7. Blau GE (1968) Generalized polynomial programming: extensions and applications. Unpublished Ph.D. Thesis, Stanford University, Dept. of Chemical Engineering

  8. Bruni L (1964) Internal economies of scale with a given technique. J Ind. Econ 12:175–190

    Article  Google Scholar 

  9. Dembo RS (1979) Second order algorithms for the posynomial geometric programming dual, part I: Analysis. Math Prog 17:156–175

    Article  Google Scholar 

  10. Dinkel JJ, Kochenberger GA (1977) On sensitivity analysis in geometric programming. Oper Res 25:155–162

    Article  Google Scholar 

  11. Dinkel JJ, Kochenberger GA, McCarl M (1974) An approach to the numerical solutions of geometric programs. Math Prog 7:181–190

    Article  Google Scholar 

  12. Dinkel JJ, Kochenberger GA, Seppala Y (1973) On the solution of regional planning models via geometric programming. Environ Plan 5:397–408

    Article  Google Scholar 

  13. Duffin RJ (1962): Cost minimization problems treated by geometric means. Oper Res 10:668–675

    Article  Google Scholar 

  14. Duffin RJ, Peterson EL, Zener C (1967) Geometric programming — theory and application. John Wiley & Sons, New York London

    Google Scholar 

  15. Ecker JG (1972) Decomposition in separable geometric programming. JOTA 9:176–186

    Article  Google Scholar 

  16. Halter AN, Carter HO, Hocking JG (1957) A note on transcendental production functions. J Farm Econ 39:966–974

    Article  Google Scholar 

  17. Hamala M (1968) Geometric programming in terms of conjugate functions. Center for Operations Research and Econometrics, Université Catholique de Louvain Discussion Paper No. 6811, Louvain, Belgium

    Google Scholar 

  18. Hivner WA, Mehta RP (1977) Optimizing computer performance with geometric programming. Eur J Oper Res 1: 103–111

    Article  Google Scholar 

  19. Heyman M, Avriel M (1969) On a decomposition for a special class of geometric programming problems. JOTA 3:392–409

    Article  Google Scholar 

  20. Intriligator MD (1971) Mathematical optimization and economic theory. Prentice-Hall, Englewood Cliffs, NJ

    Google Scholar 

  21. Optimization Theory Appl. 26:1–146 (1978)

  22. Optimization Theory Appl 26:147–350 (1978)

  23. Kadas AS (1975) An application of geometric programming to regional economics. Papers of the Reg. Sci. Assoc. 34: 95–106

    Article  Google Scholar 

  24. Kuester JK, Mize JH (1973) Optimization techniques with fortran. McGraw Hill, New York London

    Google Scholar 

  25. Lidor G, Wilde DJ (1978) Transcendental geometric programs. JOTA 26:77–96

    Article  Google Scholar 

  26. Luenberger DG (1973) Introduction to linear and nonlinear programming. Addison-Wesley, Reading, MA

    Google Scholar 

  27. Luptáčik M (1977) Geometrische Programmierung und ökonomische Analyse. Anton Hain, Meisenheim am Glan

    Google Scholar 

  28. Luptáčik M (1979) An open input-output model with continuous substitution between primary factors as a problem of geometric programming. In: Iracki K, Malanowski K, Walukiewicz S (eds) Proceedings of the 9th IFIP Conference on Optimization Techniques. Springer, Berlin Heidelberg New York

    Google Scholar 

  29. Luptáčik M (1980) Nichtlineare Programmierung mit ökonomischen Anwendungen. Athenäum Verlag, Königstein/Ts

    Google Scholar 

  30. Mastenbroek AP, Nijkamp P (1976) A spatial environmental model for an optimal allocation of investments. In: Nijkamp P, (ed) Environmental economics vol. 2: Methods. Martinus Nijhoff Social Sciences Division, Leiden

    Google Scholar 

  31. Nicolleti B, Mariani L (1977) Generalized polynomial programming. Its approach to the solution of some management science problems. Eur J Oper Res 1:239–246

    Article  Google Scholar 

  32. Nijkamp P (1972) Planning of industrial complexes by means of geometric programming. University Press, Rotterdam

    Google Scholar 

  33. Nijkamp P, Paelinck JHP (1973) An interregional model of environmental choice. Papers of the Reg. Sci. Assoc. 31: 51–71

    Article  Google Scholar 

  34. Passy U, Wilde DJ (1967) Generalized polynomial optimization. SIAM J Appl Math 15:1344–1356

    Article  Google Scholar 

  35. Passy U, Wilde DJ (1969) Mass action and polynomial optimization. J Eng Math 3:325

    Article  Google Scholar 

  36. Peterson EL (1973) Geometric programming and some of its extensions. In [2]. pp 228–289

    Google Scholar 

  37. Peterson EL (1973) The decomposition of large (generalized) geometric programming by tearing. In: Himmelblau DM (ed) Decomposition of large-scale problems. North-Holland, Amsterdam London

    Google Scholar 

  38. Pinney W. Modi J (1974) An application of geometric programming to the capital budgeting problem. San Juan, Puerto Rico, 46. ORSA Meeting

  39. Rijckaert MJ (1973) Engineering applications of geometric programming. In [2], pp 196–220

    Google Scholar 

  40. Rijckaert MJ (1974) Sensitivity analysis in geometric programming. In: Van Moeske P (ed) Mathematical programs for activity analysis. North-Holland, Amsterdam London, pp 61–71

    Google Scholar 

  41. Rijckaert MJ, Hellinckx LJ (1973) On the application of decomposition techniques in geometric programming. In: Himmelblau DM (ed) Decomposition of large-scale problems. North-Holland, Amsterdam London

    Google Scholar 

  42. Sengupta JK, Portillo-Campbell JH (1972) The approach of geometric programming with economic applications. Z ges Staatswissen 128:437–455

    Google Scholar 

  43. Sengupta JK, Fox KA (1971) Optimization techniques in quantitative economic models. North-Holland, Amsterdam London

    Google Scholar 

  44. Theil H (1972) Substitution effects in geometric programming. Manag Sci 19:25–30

    Article  Google Scholar 

  45. Wiebking R (1978) Economic dispatch of thermal power systems using geometric programming. Eur J Oper Res 2:168–171

    Article  Google Scholar 

  46. Wiebking R (1977) Optimal engineering design under uncertainty by geometric programming. Manag Sci 23:644–651

    Article  Google Scholar 

  47. Wilde DJ (1978) Globally optimal design. John Wiley & Sons, New York

    Google Scholar 

  48. Wilde DJ, Beightler RC (1967) Foundations of optimization. Prentice-Hall, Englewood Cliffs, NJ

    Google Scholar 

  49. Zangwill W (1967) The convex simplex method. Manag Sci 14:221–238

    Article  Google Scholar 

  50. Zener C (1961) Mathematical aid in optimizing engineering designs. Proc Math Acad Sci 47:537–539

    Article  Google Scholar 

  51. Zimmermann HJ (1971) Einführung in die Grundlagen des Operations Research. Verlag Moderne Industrie, München

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Unternehmensforschung, Technische Universität Wien, Argentinierstraße 8, A-1040, Wien, Österreich

    M. Luptáčik

Authors
  1. M. Luptáčik
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luptáčik, M. Geometric programming. OR Spektrum 2, 129–143 (1981). https://doi.org/10.1007/BF01719855

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01719855

Keywords

Search

Navigation