Skip to main content
SpringerLink
Log in

Duality in generalized linear fractional programming

  • Published:
Mathematical Programming Submit manuscript

A Notice to this article was published on 01 June 1984

Abstract

We consider a generalization of a linear fractional program where the maximum of finitely many linear ratios is to be minimized subject to linear constraints. For this Min-Max problem, a dual in the form of a Max-Min problem is introduced and duality relations are established.

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. D.J. Ashton and D.R. Atkins, “Multicriteria programming for financial planning”,Journal of the Operational Research Society 30 (1979) 259–270.

    Article  MATH  Google Scholar 

  2. I. Barrodale, “Best rational approximation and strict quasi-convexity”,SIAM Journal of Numerical Analysis 10 (1973) 8–12.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Charnes and W.W. Cooper, “Programming with linear fractional functionals”,Naval Research Logistics Quarterly 9 (1962) 181–186.

    MATH  MathSciNet  Google Scholar 

  4. A. Charnes and W.W. Cooper, “Goal programming and multi-objective optimization (Part I)”,European Journal of Operational Research 1 (1977) 39–54.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Charnes, L. Cox and M. Lane, “A Note on the redesigning of a rate structure for allocation of state funds to educational institutions”, Working Paper 70-49 of Project GUM, University of Texas at Austin, (Austin, Texas, 1970).

    Google Scholar 

  6. J.P. Crouzeix, “Contributions à l'étude des fonctions quasiconvexes”, Doctoral Thesis, Université de Clermont (Clermont, France, 1977).

    Google Scholar 

  7. J.P. Crouzeix, “A duality framework in quasiconvex programming”, in: S. Schaible and W.T. Ziemba, eds.,Generalized Concavity in Optimization and Economics (Academic Press, New York, 1981) pp. 207–225.

    Google Scholar 

  8. W. Dinkelbach, “On nonlinear fractional programming”,Management Science 13 (1967) 492–498.

    MathSciNet  Google Scholar 

  9. J. Flachs, “Global saddle-point duality for quasi-concave programs, II”,Mathematical Programming 24 (1982) 326–345.

    Article  MATH  MathSciNet  Google Scholar 

  10. E.G. Gol'stein,Theory of convex progranming, Translations of mathematical monographs 36 (American Mathematical Society, Providence, Rhode Island, 1972).

    Google Scholar 

  11. R. Jagannathan, “On some properties of programming problems in parametric form pertaining to fractional programming”,Management Science 12 (1966) 609–615.

    MathSciNet  Google Scholar 

  12. R. Jagannathan and S. Schaible, “Duality in generalized fractional programming via Farkas Lemma”,Journal of Optimization Theory and Applications, to appear.

  13. J.S.H. Kornbluth, “A survey of goal programming”,OMEGA 1 (1973) 193–205.

    Article  Google Scholar 

  14. O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969).

    MATH  Google Scholar 

  15. U. Passy and A. Keslassy, “Pseudo duality and duality for explicitly quasiconvex functions”, Mimeograph Series No. 249. Faculty of Industrial Engineering and Management, Technion (Haifa, Israel, 1979).

    Google Scholar 

  16. G.S. Rubinshtein, “Duality in mathematical programming and some problems of convex analysis”, (English translation)Russian mathematical surveys 25 (1970) 171–200.

    Article  Google Scholar 

  17. S. Schaible, “Fractional programming: transformations, duality and algorithmic aspects”, Technical Report 73-9, Department of Operations Research, Stanford University (Stanford, CA, 1973).

    Google Scholar 

  18. S. Schaible, “Fractional programming I, Duality”,Management Science 22 (1976) 858–867.

    MATH  MathSciNet  Google Scholar 

  19. S. Schaible, “Duality in fractional programming: a unified approach”,Operations Research 24 (1976) 452–461.

    Article  MATH  MathSciNet  Google Scholar 

  20. S. Schaible,Analyse und Anwendungen von Quotientenprogrammen (Hain-Verlag, Meisenheim, 1978).

    MATH  Google Scholar 

  21. S. Schaible, “A survey of fractional programming”, in: S. Schaible and W.T. Ziemba, eds.,Generalized concavity in optimization and economics (Academic Press, New York, 1981) pp. 417–440.

    Google Scholar 

  22. S. Schaible, “Bibliography in fractional programming”,Zeitschrift für Operations Research 26 (7) (1982).

  23. E.C. Tammer, “Dualitätstheorie für hyperbolische und stückweise-lineare konvexe Optimierungs-probleme”,Mathematische Operationenforschung und Statistik 5 (1974) 93–108.

    MathSciNet  Google Scholar 

  24. R. Vogt, “A corporate strategy for realizing equal employment opportunity”,Behavioral and Social Accounting, to appear.

  25. J. von Neumann, “A model of general economic equilibrium”,Review of Economic Studies 13 (1945) 1–9.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de mathématiques appliquées, Université de Clermont, Aubière, France

    Jean-Pierre Crouzeix

  2. Départment d'informatique et de recherche opérationelle, Université de Montréal, Montréal, PQ, Canada

    Jacques A. Ferland

  3. Department of Finance and Management Science, University of Alberta, Edmonton, AB, Canada

    Siegfried Schaible

Authors
  1. Jean-Pierre Crouzeix
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacques A. Ferland
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siegfried Schaible
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by a Research Grant from NATO, No. 1934.

Additionally supported by grant NSERC, A8312.

Additionally supported by grant NSERC, A4534.

An erratum to this article is available at http://dx.doi.org/10.1007/BF02592224.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Crouzeix, JP., Ferland, J.A. & Schaible, S. Duality in generalized linear fractional programming. Mathematical Programming 27, 342–354 (1983). https://doi.org/10.1007/BF02591908

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02591908

Key words

Search

Navigation