caa a22 4500 475769139 CHVBK 20180406123615.0 cr unu---uuuuu 170329e20000101xx s 000 0 eng 10.1007/s101079900113 doi (NATIONALLICENCE)springer-10.1007/s101079900113 Forcing strong convergence of proximal point iterations in a Hilbert space [Elektronische Daten] [M.V. Solodov, B.F. Svaiter] Abstract. : This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by Güler [11] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns. Springer-Verlag Berlin Heidelberg, 2000 Key words: proximal point algorithm - Hilbert spaces - weak convergence - strong convergence Mathematics Subject Classification (1991): 49M45, 90C25, 90C33 nationallicence Solodov M.V. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil. e-mail: solodov@impa.br,benar@impa.br, BR aut Svaiter B.F. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil. e-mail: solodov@impa.br,benar@impa.br, BR aut https://doi.org/10.1007/s101079900113 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s101079900113 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Solodov M.V. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil. e-mail: solodov@impa.br,benar@impa.br, BR aut NATIONALLICENCE 700 1- Svaiter B.F. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil. e-mail: solodov@impa.br,benar@impa.br, BR aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer