# Convergence of Prox-Regularization Methods for Generalized Fractional Programming

RAIRO - Operations Research (2010)

- Volume: 36, Issue: 1, page 73-94
- ISSN: 0399-0559

## Access Full Article

top## Abstract

top## How to cite

topRoubi, Ahmed. "Convergence of Prox-Regularization Methods for Generalized Fractional Programming." RAIRO - Operations Research 36.1 (2010): 73-94. <http://eudml.org/doc/105262>.

@article{Roubi2010,

abstract = {
We analyze the convergence of the prox-regularization algorithms
introduced in [1], to solve generalized fractional programs,
without assuming that the optimal solutions set of the considered
problem is nonempty, and since the objective functions are
variable with respect to the iterations in the auxiliary problems
generated by Dinkelbach-type algorithms DT1 and DT2, we consider
that the regularizing parameter is also variable. On the other
hand we study the convergence when the iterates are only
ηk-minimizers of the auxiliary problems. This situation is
more general than the one considered in [1]. We also give some
results concerning the rate of convergence of these algorithms,
and show that it is linear and some times superlinear for some
classes of functions. Illustrations by numerical examples are
given in [1].
},

author = {Roubi, Ahmed},

journal = {RAIRO - Operations Research},

keywords = {Generalized fractional programs; Dinkelbach-type
algorithms; proximal point algorithm;
rate of convergence.; Generalized fractional programs, Dinkelbach-type algorithms, proximal point algorithm, rate of convergence.},

language = {eng},

month = {3},

number = {1},

pages = {73-94},

publisher = {EDP Sciences},

title = {Convergence of Prox-Regularization Methods for Generalized Fractional Programming},

url = {http://eudml.org/doc/105262},

volume = {36},

year = {2010},

}

TY - JOUR

AU - Roubi, Ahmed

TI - Convergence of Prox-Regularization Methods for Generalized Fractional Programming

JO - RAIRO - Operations Research

DA - 2010/3//

PB - EDP Sciences

VL - 36

IS - 1

SP - 73

EP - 94

AB -
We analyze the convergence of the prox-regularization algorithms
introduced in [1], to solve generalized fractional programs,
without assuming that the optimal solutions set of the considered
problem is nonempty, and since the objective functions are
variable with respect to the iterations in the auxiliary problems
generated by Dinkelbach-type algorithms DT1 and DT2, we consider
that the regularizing parameter is also variable. On the other
hand we study the convergence when the iterates are only
ηk-minimizers of the auxiliary problems. This situation is
more general than the one considered in [1]. We also give some
results concerning the rate of convergence of these algorithms,
and show that it is linear and some times superlinear for some
classes of functions. Illustrations by numerical examples are
given in [1].

LA - eng

KW - Generalized fractional programs; Dinkelbach-type
algorithms; proximal point algorithm;
rate of convergence.; Generalized fractional programs, Dinkelbach-type algorithms, proximal point algorithm, rate of convergence.

UR - http://eudml.org/doc/105262

ER -

## References

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.