Abstract
The notion of lower subdifferentiability is applied to the analysis of convex fractional programming problems. In particular, duality results and optimality conditions are presented, and the applicability of a cutting-plane algorithm using lower subgradients is discussed. These methods are useful also in generalized fractional programming, where, in the linear case, the performance of the cutting-plane algorithm is compared with that of the most efficient version of the Dinkelbach method, which is based on the solution of a parametric linear programming problem.
Similar content being viewed by others
References
Schaible, S., andIbaraki, T.,Fractional Programming, European Journal of Operational Research, Vol. 12, pp. 325–338, 1983.
Craven, B. D.,Fractional Programming, Heldermann-Verlag, Berlin, Germany, 1988.
Lasdon, L. S.,Optimization Theory for Large Systems, Macmillan, London, England, 1970.
Barrodale, I.,Best Rational Approximation and Strict Quasiconvexity, SIAM Journal on Numerical Analysis, Vol. 10, pp. 8–12, 1973.
Schaible, S.,Fractional Programming, Zeitschrift für Operations Research, Vol. 27, pp. 39–54, 1983.
Plastria, F.,Lower Subdifferentiable Functions and Their Minimization by Cutting Planes, Journal of Optimization Theory and Applications, Vol. 46, pp. 37–53, 1985.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Penot, J. P., andVolle, M.,Another Duality Scheme for Quasiconvex Problems, Trends in Mathematical Optimization, Edited by K. H. Hoffmann, J. B. Hiriart-Urruty, C. Lemarechal, and J. Zowe, Birkhäuser-Verlag, Basel, Switzerland, pp. 259–275, 1988.
Penot, J. P., andVolle, M.,On Quasiconvex Duality, Unpublished Manuscript, 1986.
Martínez-Legaz, J. E.,On Lower Subdifferentiable Functions, Trends in Mathematical Optimization, Edited by K. H. Hoffmann, J. B. Hiriart-Urruty, C. Lemarechal, and J. Zowe, Birkhäuser-Verlag, Basel, Switzerland, pp. 197–232, 1988.
Martínez-Legaz, J. E.,Quasiconvex Duality Theory by Generalized Conjugation Methods, Optimization, Vol. 19, pp. 603–652, 1988.
Plastria, F.,The Minimization of Lower Subdifferentiable Functions under Nonlinear Constraints: An All-Feasible Cutting Plane Algorithm, Journal of Optimization Theory and Applications, Vol. 57, pp. 463–485, 1988.
Crouzeix, J. P.,About Differentiability of Order One of Quasiconvex Functions on R n, Journal of Optimization Theory and Applications, Vol. 36, pp. 367–385, 1982.
Mangasarian, O. L.,Nonlinear Programming, McGraw Hill, New York, New York, 1969.
Crouzeix, J. P.,Contributions a l'Étude des Fonctions Quasiconvexes, Université de Clermont-Ferrand II, Thèse, 1977.
Greenberg, H. P., andPierskalla, W. P.,Quasiconjugate Functions and Surrogate Duality, Cahiers du Centre d'Études de Recherche Operationelle, Vol. 15, pp. 437–448, 1973.
Crouzeix, J. P.,A Duality Framework in Quasiconvex Programming, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, pp. 207–225, 1981.
Plastria, F.,Testing Whether a Cutting-Plane May Be Dropped, Revue Belge de Statistique, d'Informatique et de Recherche Opérationnelle, Vol. 22, pp. 11–21, 1982.
Ferland, J. A., andPotvin, J. Y.,Generalized Fractional Programming: Algorithms and Numerical Experimentation, European Journal of Operational Research, Vol. 20, pp. 92–101, 1985.
Borde, J., andCrouzeix, J. P.,Convergence of a Dinkelbach-Type Algorithm in Generalized Fractional Programming, Zeitschrift für Operations Research, Vol. 31, pp. A31-A54, 1987.
Dyer, M. E.,Calculating Surrogate Constraints, Mathematical Programming, Vol. 19, pp. 255–278, 1980.
Sikorski, J.,Quasi-Subgradient Algorithms for Calculating Surrogate Constraints, Analysis and Algorithms of Optimization Problems, Edited by K. Malanowski and K. Mizukami, Springer-Verlag, Berlin, Germany, pp. 203–236, 1986.
Schroeder, R. G.,Linear Programming Solutions to Ratio Games, Operations Research, Vol. 18, pp. 300–305, 1970.
Author information
Authors and Affiliations
Additional information
Communicated by O. L. Mangasarian
The authors wish to thank Mr. Jaume Timoneda for his help in the implementation of the numerical methods on the computer and the referees for valuable comments and suggestions; the present improved statement and proof of Proposition 2.1 is due to one of them. Financial support from the Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058, is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Boncompte, M., Martínez-Legaz, J.E. Fractional programming by lower subdifferentiability techniques. J Optim Theory Appl 68, 95–116 (1991). https://doi.org/10.1007/BF00939937
Issue Date:
DOI: https://doi.org/10.1007/BF00939937