- Title
- Convergence of the proximal point method for metrically regular mappings
- Creator
- Aragón Artacho, Francisco J.; Dontchev, A. L.; Geoffroy, M. H.
- Relation
- ESAIM: Proceedings Vol. 17, Issue April, p. 1-8
- Publisher Link
- http://dx.doi.org/10.1051/proc:071701
- Publisher
- E. D. P. Sciences
- Resource Type
- journal article
- Date
- 2007
- Description
- In this paper we consider the following general version of the proximal point algorithm for solving the inclusion T(x) ∋ 0, where T is a set-valued mapping acting from a Banach space X to a Banach space Y. First, choose any sequence of functions gn : X → Y with gn(0) = 0 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 gn(xn1-xn)+T(xn+1) ∋ 0 for n = 0,1,... 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 x̅ (x̅ being a solution to T(x) ∋ 0) such that for each initial point x₀ ∈ O one can find a sequence xn generated by the algorithm which is linearly convergent to x̅. Moreover, if the functions gn have their Lipschitz constants convergent to zero, then there exists a sequence starting from x₀ ∈ O which is superlinearly convergent to x̅. Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular.
- Subject
- proximal point algorithm; set-valued mapping; metric regularity; subregularity; strong regularity; variational inequality; optimization
- Identifier
- http://hdl.handle.net/1959.13/933728
- Identifier
- uon:11701
- Identifier
- ISSN:1270-900X
- Rights
- The original publication is available at http://www.esaim‐proc.org
- Language
- eng
- Full Text
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