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Generalized Convexity, Generalized Monotonicity and Applications pp 147–159Cite as

Duality in Fractional Programming Problems with Set Constraints

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Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 77))

Abstract

Duality is studied for a minimization problem with finitely many inequality and equality constraints and a set constraint where the constraining convex set is not necessarily open or closed. Under suitable generalized convexity assumptions we derive a weak, strong and strict converse duality theorem. By means of a suitable transformation of variables these results are then applied to a class of fractional programs involving a ratio of a convex and an affine function with a set constraint in addition to inequality and equality constraints. The results extend classical fractional programming duality by allowing for a set constraint involving a convex set that is not necessarily open or closed.

This research has been partially supported by M.I.U.R. and C.N.R.

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References

  • Barros, A.I., Frenk, J.B.G., Schaible, S. and Zhang, S. (1996), Using duality to solve generalized fractional programming problems, Journal of Global Optimization, Vol. 8, pp. 139–170.

    Article  MathSciNet  MATH  Google Scholar 

  • Barros A. I. (1998), Discrete and Fractional Programming Techniques for Location Models, Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  • Bector, C.R. (1973), Duality in nonlinear fractional programming, Zeitschrift fur Operations Research, Vol. 17, pp. 183–193.

    Article  MATH  MathSciNet  Google Scholar 

  • Bector, C.R., Bector, M.H. and Klassen, J.E. (1977), Duality for a nonlinear programming problem, Utilitas Mathematicae, Vol. 11, pp. 87–99.

    MathSciNet  MATH  Google Scholar 

  • Cambini R. (1996), Some new classes of generalized concave vectorvalued functions, Optimization, Vol. 36, n. 1, pp. 11–24.

    MATH  MathSciNet  Google Scholar 

  • Cambini R. and S. Komlósi (1998), On the Scalarization of Pseudoconcavity and Pseudomonotonicity Concepts for Vector Valued Functions, in Generalized Convexity, Generalized Monotonicity: Recent Results, edited by J.-P. Crouzeix, J.-E. Martinez-Legaz and M. Volle, Nonconvex Optimization and Its Applications, Vol. 27, Kluwer Academic Publishers, Dordrecht, pp. 277–290.

    Google Scholar 

  • Cambini, R. and Carosi, L. (2002), Duality in multiobjective optimization problems with set constraints, Report n. 233, Department of Statistics and Applied Mathematics, University of Pisa.

    Google Scholar 

  • Chandra, S., Abha Goyal and Husain, I. (1998), On symmetric duality in mathematical programming with F-convexity, Optimization, Vol. 43, pp. 1–18.

    MathSciNet  MATH  Google Scholar 

  • Charnes, A. and Cooper, W.W. (1962), Programming with linear fractional functionals, Naval Research Logistic Quarterly, Vol. 9, pp. 181–196.

    MathSciNet  MATH  Google Scholar 

  • Craven, B.D. (1981), Duality for generalized convex fractional programs, in Generalized Concavity in Optimization and Economics, edited by S. Schaible and W.T. Ziemba, Academic Press, New York, pp. 473–489.

    Google Scholar 

  • Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1983), Duality in generalized linear fractional programming, Mathematical Programming, vol. 27, pp. 342–354.

    MathSciNet  MATH  Google Scholar 

  • Dinkelbach, W. (1967), On nonlinear fractional programming, Management Science, Vol. 13, pp. 492–498.

    MATH  MathSciNet  Google Scholar 

  • Frenk, J.B.G. and Kasssay, G. (1999), On classes of generalized convex functions, Gordan-Farkas type theorems and Lagrangean duality, Journal of Optimization Theory and Applications, Vol. 102, n. 2, pp. 315–343.

    Article  MathSciNet  MATH  Google Scholar 

  • Geoffrion, A.M. (1971), Duality in nonlinear programming: a simplified applications-oriented development, SIAM Review, Vol. 12, pp. 1–37.

    Article  MathSciNet  Google Scholar 

  • Giorgi, G. and Guerraggio, A. (1994), First order generalized optimality conditions for programming problems with a set constraint, in Generalized Convexity, edited by S. Komlósi, T. Rapcsák and S. Schaible, Lecture Notes in Economics and Mathematical Systems, Vol. 405, Springer-Verlag, Berlin, pp. 171–185.

    Google Scholar 

  • Jagannathan, R. (1973), Duality for nonlinear fractional programs, Zeitschrift fur Operations Research, Vol. 17, pp. 1–3

    Article  MATH  MathSciNet  Google Scholar 

  • Jahn, J. (1994), Introduction to the Theory of Nonlinear Optimization, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Liang, Z.A., Huang, H.X. and Pardalos, P.M. (2001), Optimality conditions and duality for a class of nonlinear fractional programming problems, Journal of Optimization Theory and Applications, Vol. 110, pp. 611–619.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, J.C. (1996), Optimality and duality for generalized fractional programming involving nonsmooth pseudoinvex functions, Journal of Mathematical Analysis and Applications, Vol. 202, pp. 667–685.

    Article  MATH  MathSciNet  Google Scholar 

  • Mahajan, D.G. and Vartak, M.N. (1977), Generalization of some duality theorems in nonlinear programming, Mathematical Programming, Vol. 12, pp. 293–317.

    Article  MathSciNet  MATH  Google Scholar 

  • Mangasarian, O.L. (1969), Nonlinear Programming, McGraw-Hill, N.Y.

    MATH  Google Scholar 

  • Mond, B. and Weir, T. (1981), Generalized concavity and duality, in Generalized Concavity in Optimization and Economics, edited by S. Schaible and W.T. Ziemba, Academic Press, New York, pp. 263–279.

    Google Scholar 

  • Schaible, S. (1973), Fractional programming: transformations, duality and algorithmic aspects, Technical Report 73-9, Department of Operation Research, Stanford University, November 1973.

    Google Scholar 

  • Schaible, S. (1974), Parameter-free convex equivalent and dual programs of fractional programming problems, Zeitschrift fur Operations Research, Vol. 18, pp. 187–196.

    Article  MATH  MathSciNet  Google Scholar 

  • Schaible, S. (1976), Duality in fractional programming: a unified approach, Operations Research, Vol. 24, pp. 452–461.

    Article  MATH  MathSciNet  Google Scholar 

  • Schaible, S. (1976), Fractional programming. I, duality, Management Science, Vol. 22, pp. 858–867.

    MATH  MathSciNet  Google Scholar 

  • Scott, C.H. and Jefferson, T.R. (1996), Convex dual for quadratic concave fractional programs, Journal of Optimization Theory and Applications, Vol. 91, pp. 115–122.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Statistics and Applied Mathematics, University of Pisa, Italy

    Riccardo Cambini & Laura Carosi

  2. A. Gary Anderson Graduate School of Management, University of California at Riverside, USA

    Siegfried Schaible

Authors
  1. Riccardo Cambini
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  2. Laura Carosi
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  3. Siegfried Schaible
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Editor information

Editors and Affiliations

  1. RMIT University, Australia

    Andrew Eberhard

  2. University of the Aegean, Greece

    Nicolas Hadjisavvas

  3. University of Avignon, France

    Dinh The Luc

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© 2005 Springer Science + Business Media, Inc.

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Cambini, R., Carosi, L., Schaible, S. (2005). Duality in Fractional Programming Problems with Set Constraints. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol 77. Springer, Boston, MA. https://doi.org/10.1007/0-387-23639-2_8

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