Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-09T07:24:11.518Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of Prox-Regularization Methods for Generalized Fractional Programming

Published online by Cambridge University Press:  15 July 2002

Ahmed Roubi
Ahmed Roubi*
Affiliation:
Faculté des Sciences et Techniques, Département de Mathématiques et Informatique, Route de Casablanca, BP. 577, Settat, Morocco; [email protected].
Get access

Abstract

We analyze the convergence of the prox-regularization algorithmsintroduced in [1], to solve generalized fractional programs,without assuming that the optimal solutions set of the consideredproblem is nonempty, and since the objective functions arevariable with respect to the iterations in the auxiliary problemsgenerated by Dinkelbach-type algorithms DT1 and DT2, we considerthat the regularizing parameter is also variable. On the otherhand we study the convergence when the iterates are onlyηk -minimizers of the auxiliary problems. This situation ismore general than the one considered in [1]. We also give someresults concerning the rate of convergence of these algorithms,and show that it is linear and some times superlinear for someclasses of functions. Illustrations by numerical examples aregiven in [1].

Keywords

Generalized fractional programsDinkelbach-type algorithmsproximal point algorithmrate of convergence.
Type
Research Article
Information
RAIRO - Operations Research , Volume 36 , Issue 1 , January 2002 , pp. 73 - 94
Copyright
© EDP Sciences, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gugat, M., Prox-Regularization Methods for Generalized Fractional Programming. J. Optim. Theory Appl. 99 (1998) 691-722. CrossRef
Crouzeix, J.-P., Ferland, J.A. and Schaible, S., Algorithm, An for Generalized Fractional Programs. J. Optim. Theory Appl. 47 (1985) 35-49. CrossRef
Crouzeix, J.-P., Ferland, J.A. and Schaible, S., Note on an Algorithm for Generalized Fractional Programs. J. Optim. Theory Appl. 50 (1986) 183-187. CrossRef
Dinkelbach, W., Nonlinear Fractional Programming, On. Management Sci. 13 (1967) 492-498. CrossRef
Roubi, A., Method of Centers for Generalized Fractional Programming. J. Optim. Theory Appl. 107 (2000) 123-143. CrossRef
Martinet, B., Régularisation d'Inéquations Variationnelles par Approximation Successives. RAIRO: Oper. Res. 4 (1970) 154-158. CrossRef
Rockafellar, R.T., Monotone Operators and the Proximal Point Algorithm. SIAM J. Control Optim. 14 (1976) 877-898. CrossRef
Güler, O., On the Convergence of the Proximal Point Algorithm for Convex Minimization. SIAM J. Control Optim. 29 (1991) 403-419. CrossRef
A. Kaplan and R. Tichatschke, Stable Methods for Ill-Posed Variational Problems. Akademic Verlag, Berlin, Germany (1994).
Lemaréchal, C. and Sagastizábal, C., Practical Aspects of the Moreau-Yosida Regularization: Theoretical Preliminaries. SIAM J. Optim. 7 (1997) 367-385. CrossRef
I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels. Gauthier-Villars, Paris, Bruxelles, Montréal (1974).
Burke, J.V. and Ferris, M.C., Weak Sharp Minima in Mathematical Programming. SIAM J. Control Optim. 31 (1993) 1340-1359. CrossRef
Cornejo, O., Jourani, A. and Zalinescu, C., Conditioning and Upper-Lipschitz Inverse Subdifferentials in Nonsmooth Optimization Problems. J. Optim. Theory Appl. 95 (1997) 127-148. CrossRef