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Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.13/933728
- Convergence of the proximal point method for metrically regular mappings
Aragón Artacho, Francisco J.;
Dontchev, A. L.;
Geoffroy, M. H.
- 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.
- ESAIM: Proceedings Vol. 17, Issue April, p. 1-8
- Publisher Link
- E. D. P. Sciences
proximal point algorithm;
- Resource Type
- journal article
- The original publication is available at http://www.esaim‐proc.org