Convergence of Prox-Regularization Methods for Generalized Fractional Programming

Ahmed Roubi

RAIRO - Operations Research (2010)

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

Abstract

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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].

How to cite

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Roubi, 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

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  9. A. Kaplan and R. Tichatschke, Stable Methods for Ill-Posed Variational Problems. Akademic Verlag, Berlin, Germany (1994).  Zbl0804.49011
  10. C. Lemaréchal and C. Sagastizábal, Practical Aspects of the Moreau-Yosida Regularization: Theoretical Preliminaries. SIAM J. Optim.7 (1997) 367-385.  Zbl0876.49019
  11. I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels. Gauthier-Villars, Paris, Bruxelles, Montréal (1974).  
  12. J.V. Burke and M.C. Ferris, Weak Sharp Minima in Mathematical Programming. SIAM J. Control Optim.31 (1993) 1340-1359.  Zbl0791.90040
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