Abstract
In generalized fractional programming, one seeks to minimize the maximum of a finite number of ratios. Such programs are, in general, nonconvex and consequently are difficult to solve. Here, we consider a particular case in which the ratio is the quotient of a quadratic form and a positive concave function. The dual of such a problem is constructed and a numerical example is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avriel, M., Diewert, W., Schaible, S. and Zang, I. (1988), Generalized Concavity. Plenum Press.
Barros, A. I. (1995), Discrete and fractional programming techniques for location models, Tinbergen Institute Research Series 89.
Barros, A. I., Frenk, J. B. G., Schaible, S. and Zhang, S. (1996a), Using duality to solve generalized fractional programming problems, Journal of Global Optimization, 8, 139–170.
Barros, A. I., Frenk, J. B. G., Schaible, S. Zhang, S., (1996b), Anew algorithm for generalized fractional programs, Mathematical Programming 72, 147–175.
Bector, C. R. (1968), Programming problems with convex fractional functions, Operations Research 16, 383–391.
Bector, C. R., and Suneja, S. K. (1988), Duality in nondifferentiable generalized fractional programming, {tiAsia Pacific Journal of Operational Research} 5(2), 134–139.
Boncompte, M., and Martinez-Legaz, J. E. (1991), Fractional programming by lower subdifferentiability techniques, Journal of Optimization Theory and Applications 68(1), 95–116.
Chandra, S., Craven, B. D., and Mond, B. (1986), Generalized fractional programming duality: A ratio game approach, Australian Mathematical Society, Journal Series B, Applied Mathematics 28(2), 170–180.
Crouzeix, J. P., Ferland, J. A., and Schaible, S. (1983), Duality in generalized linear fractional programming, {tiMathematical Programming} 27(3), 342–354.
Hiriart-Urruty, J. B., and Lemarechal, C. (1993), Convex Analysis and Minimization Algorithms. Springer Verlag.
Jagannathan, R., and Schaible, S. (1983), Duality in generalized fractional programming via Farkas' Lemma, Journal of Optimization Theory and Applications 41(3), 417–424.
Jefferson, T. R. and C. H. Scott (1978), Avenues of geometric programming I. Theory, New Zealand Operational Research, 6, 109–136.
Martein, L., and Sodini, C. (1982), Un Algoritmo per un problema di programmazione frazionaria non lineare e non convessa, Publication No. 93, Serie A, Dipartimento di Ricerca Operativa e Scienze Statistiche, Università di Pisa, Italy.
Peterson, E. L. (1976), Geometric Programming, SIAM Review 18, 1–52.
Rockafellar, R. T. (1970), Convex Analysis. Princeton University Press, Princeton, New Jersey.
Schaible, S. (1995), Fractional programming, in R. Horst and P. M. Pardalos (eds.), Handbook of Global Optimization, Kluwer, 495–608.
Scott, C. H., and Jefferson, T. R. (1989), Conjugate duality in generalized fractional programming, {tiJournal of Optimization Theory and Applications} 60(3), 475–483.
Scott, C. H. and Jefferson, T. R. (1996), A convex dual for quadratic-concave fractional programs, {tiJournal of Optimization, Theory and Applications}, to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Scott, C.H., Jefferson, T.R. & Frenk, J.B.G. A Duality Theory for a Class of Generalized Fractional Programs. Journal of Global Optimization 12, 239–245 (1998). https://doi.org/10.1023/A:1008274708071
Issue Date:
DOI: https://doi.org/10.1023/A:1008274708071