Geodesic
Convergence of the Proximal Point Method for Metrically Regular Mappings
F. J. Aragón Artacho1 ; A. L. Dontchev2 ; M. H. Geoffroy3
1 Department of Statistics and Operations Research, University of Alicante, 03071 Alicante, Spain, This author is supported by Grant BES-2003-0188 from FPI Program of MEC (Spain).
2 Mathematical Reviews, Ann Arbor, MI 48107, USA, .
3 Laboratoire AOC, Dpt. de Mathématiques, Université Antilles-Guyane, F-97159 Pointe-à-Pitre, Guadeloupe, . This author is supported by Contract EA3591 (France).
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In this paper we consider the following general version of the proximal point algorithm for solving the inclusion , where T is a set-valued mapping acting from a Banach space X to a Banach space Y. First, choose any sequence of functions with that are Lipschitz continuous in a neighborhood of the origin. Then pick an initial guess x0 and find a sequence xn by applying the iteration for We prove that if the Lipschitz constants of gn are bounded by half the reciprocal of the modulus of regularity of T, then there exists a neighborhood O of ( being a solution to ) such that for each initial point one can find a sequence xn generated by the algorithm which is linearly convergent to . Moreover, if the functions gn have their Lipschitz constants convergent to zero, then there exists a sequence starting from which is superlinearly convergent to . Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular.
DOI : 10.1051/proc:071701

F. J. Aragón Artacho 1 ; A. L. Dontchev 2 ; M. H. Geoffroy 3

1 Department of Statistics and Operations Research, University of Alicante, 03071 Alicante, Spain, This author is supported by Grant BES-2003-0188 from FPI Program of MEC (Spain).
2 Mathematical Reviews, Ann Arbor, MI 48107, USA, .
3 Laboratoire AOC, Dpt. de Mathématiques, Université Antilles-Guyane, F-97159 Pointe-à-Pitre, Guadeloupe, . This author is supported by Contract EA3591 (France). 
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     author = {F. J. Arag\'on Artacho and A. L. Dontchev and M. H. Geoffroy},
     title = {Convergence of the {Proximal} {Point} {Method} for {Metrically} {Regular} {Mappings}},
     journal = {ESAIM. Proceedings},
     pages = {1--8},
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     volume = {17},
     doi = {10.1051/proc:071701},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc:071701/}
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F. J. Aragón Artacho; A. L. Dontchev; M. H. Geoffroy. Convergence of the Proximal Point Method for Metrically Regular Mappings. ESAIM. Proceedings, Tome 17 (2007), pp. 1-8. doi: 10.1051/proc:071701

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