On an algorithm in nondifferential convex optimization
DOI:
https://doi.org/10.2298/YJOR110501024DKeywords:
Moreau-Yosida regularization, non-smooth convex optimization, directional derivative, second order Dini upper directional derivative, uniformly convex functionsAbstract
In this paper an algorithm for minimization of a nondifferentiable function is presented. The algorithm uses the Moreau-Yosida regularization of the objective function and its second order Dini upper directional derivative. The purpose of the paper is to establish general hypotheses for this algorithm, under which convergence occurs to optimal points. A convergence proof is given, as well as an estimate of the rate of the convergence.References
Clarke F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
Djuranovic Milicic Nada: “A generalization of the Cea-Goldstein step-size algorithm”, European Journal of Operational Research 63 (1992) 1-9.
Djuranovic Milicic Nada: “On an algorithm for C1,1 optimization”, Filomat, 21 (2007) 17-24.
Djuranovic-Milicic, N., “An algorithm for LC1 optimization”, YUJOR, 15 (2) (2005) 301-300.
Djuranovic-Milicic, N., “A multi-step curve search algorithm in nonlinear optimization”, YUJOR, 18 (1) (2008) 47-52.
Djuranovic-Milicic, N., and Gardasevic-Filipovic M.: “An algorithm for minimization of a nondifferentiable convex function“, The 2009 International Conference of Applied and Engineering Mathematics, World Congress on Engineering 2009, Proceedings, 2004, 1241-1246.
Демьянов В. Ф., Васильев: Недифференцируемая оптимизация, Москва „Наука“, 1981.
Fuduli A.: “Metodi numerici per la minimizzazione di funzioni convesse nondifferenziabili”, PhD Thesis, Departimento di Electronica Informatica e Sistemistica, Univerzita della Calabria, 1998.
Fletcher R.: Practical Methods of Optimization, Volume 2, Constrained Optimization, Department of Mathematics, University of Dundee, Scotland, U.K, Wiley-Interscience, Publication, 1981.
Иоффе А. Д., Тихомиров В. М.: Теория экстремальных задач, Москва „Наука“, 1974.
Karmanov V.G: “Mathematical Programming”. Nauka, Moscow, 1975 (in Russian).
Кусарев А. Г., Кутателадзе С. С.: Субдифференциальное исчисление, Новосибирск, Издательство „Наука“, 1987.
Lemaréchal C., and Sagastizabal C.: “Practical aspects of the Moreau-Yosida regularization I: theoretical preliminaries”, SIAM Journal of Optimization, 7 (1997) 367-385.
Meng F., and Zhao G.: ”On second-order properties of the Moreau-Yosida regularization for constrained nonsmooth convex programs”, Numer. Funct. Anal. Optim, VOL 25; NUMB 5/6, pages 515-530, 2004.
Пшеничный Б. Н.: Выпуклый анализ и экстремальные задачи, Москва „Наука“, 1980.
Пшеничный Б. Н.: Необходимые условия экстремума, Москва „Наука“, 1982.
Пшеничный Б. Н.: Метод линеаризации, Москва „Наука“, 1983.
Rockafellar R.: Convex Analysis, Princeton University Press, New Jersey, 1970.
Polak E.: Computational Methods in Optimization. A Unified Approach, Mathematics in Science and Engineering, Vol. 77, Academic Press, New York - London, 1971.
Pytlak R.: ”Conjugate gradient algorithms in nonconvex optimization”, Library of Congress Control Number: 2008937497, Springer-Verlag, Berlin Heidelberg, 2009.
Shi, Z.J., “Convergence of multi-step curve search method for unconstrained optimization”, J. Numer. Math., 12 (2004) 297-309.
Sun W., Sampaio R.J.B., Yuan J.: “Two algorithms for LC unconstrained optimization”, Journal of Computational Mathematics, 18 (2000) 621-632.
Qi L.: “Second order analysis of the Moreau-Yosida regularization”, School of Mathematics, The University of New Wales Sydney, New South Wales 2052, Australia, 1998.
Downloads
Published
Issue
Section
License
Copyright (c) 2013 YUJOR
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
