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On the Pseudoconvexity of the Sum of Two Linear Fractional Functions

  • Alberto Cambini
  • Laura Martein
  • Siegfried Schaible
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 77)

Abstract

Charnes and Cooper (1962) reduced a linear fractional program to a linear program with help of a suitable transformation of variables. We show that this transformation preserves pseudoconvexity of a function. The result is then used to characterize sums of two linear fractional functions which are still pseudoconvex. This in turn leads to a characterization of pseudolinear sums of two linear fractional functions.

Keywords

Fractional programming sum of ratios pseudoconvexity pseudolinearity 

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References

  1. Avriel M., Diewert W. E., Schaible S. and Zang I., Generalized concavity, Plenum Press, New York, 1988.Google Scholar
  2. Cambini A. and Martein L., A modified version of Martos’s algorithm for the linear fractional problem, Methods of Operations Research, 53, 1986, 33–44.MathSciNetGoogle Scholar
  3. Cambini A., Crouzeix J.P. and Martein L., On the pseudoconvexity of a quadratic fractional function, Optimization, 2002, vol. 51(4), 677–687.MathSciNetCrossRefGoogle Scholar
  4. Cambini A., Martein L. and Schaible S., On maximizing a sum of ratios, J. of Information and Optimization Sciences, 10, 1989, 65–79.MathSciNetGoogle Scholar
  5. Cambini A. and Martein L., Generalized concavity and optimality conditions in vector and scalar optimization, “Generalized convexity” (Komlosi et al. eds.), Lect. Notes Econom. Math. Syst., 405, Springer-Verlag, Berlin, 1994, 337–357.Google Scholar
  6. Cambini R. and Carosi L., On generalized convexity of quadratic fractional functions, Technical Report n.213, Dept. of Statistics and Applied Mathematics, University of Pisa, 2001.Google Scholar
  7. Charnes A. and Cooper W. W., Programming with linear fractional functionals, Nav. Res. Logist. Quart., 9, 1962, 181–196.MathSciNetGoogle Scholar
  8. Craven B. D., Fractional programming, Sigma Ser. Appl. Math. 4, Heldermann Verlag, Berlin, 1988.Google Scholar
  9. Crouzeix J.P., Characterizations of generalized convexity and monotonicity, a survey, Generalized convexity, generalized monotonicity (Crouzeix et al. eds.), Kluwer Academic Publisher, Dordrecht, 1998, 237–256.Google Scholar
  10. Jeyakumar V. and Yang X. Q., On characterizing the solution sets of pseudolinear programs, J. Optimization Theory Appl., 87, 1995, 747–755.MathSciNetCrossRefGoogle Scholar
  11. Komlosi S., First and second-order characterization of pseudolinear functions, Eur, J. Oper. Res., 67, 1993, 278–286.zbMATHCrossRefGoogle Scholar
  12. Martos B., Nonlinear programming theory and methods, North-Holland, Amsterdam, 1975.Google Scholar
  13. Rapcsak T., On pseudolinear functions, Eur. J. Oper. Res., 50, 1991, 353–360.zbMATHCrossRefGoogle Scholar
  14. Schaible S., A note on the sum of a linear and linear-fractional function, Nav. Res. Logist. Quart., 24, 1977, 691–693.zbMATHGoogle Scholar
  15. Schaible S., Fractional programming, Handbook of global optimization (Horst and Pardalos eds.), Kluwer Academic Publishers, Dordrecht, 1995, 495–608.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Alberto Cambini
    • 1
  • Laura Martein
    • 1
  • Siegfried Schaible
    • 2
  1. 1.Department of Statistics and Applied MathematicsUniversity of PisaItaly
  2. 2.A. G. Anderson Graduate School of ManagementUniversity of California at RiversideUSA

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