caa a22 4500 475769732 CHVBK 20180406123616.0 cr unu---uuuuu 170329e20000801xx s 000 0 eng 10.1007/s101070050022 doi (NATIONALLICENCE)springer-10.1007/s101070050022 Error bounds for proximal point subproblems and associated inexact proximal point algorithms [Elektronische Daten] [M.V. Solodov, B.F. Svaiter] Abstract. : We study various error measures for approximate solution of proximal point regularizations of the variational inequality problem, and of the closely related problem of finding a zero of a maximal monotone operator. A new merit function is proposed for proximal point subproblems associated with the latter. This merit function is based on Burachik-Iusem-Svaiter's concept of ε-enlargement of a maximal monotone operator. For variational inequalities, we establish a precise relationship between the regularized gap function, which is a natural error measure in this context, and our new merit function. Some error bounds are derived using both merit functions for the corresponding formulations of the proximal subproblem. We further use the regularized gap function to devise a new inexact proximal point algorithm for solving monotone variational inequalities. This inexact proximal point method preserves all the desirable global and local convergence properties of the classical exact/inexact method, while providing a constructive error tolerance criterion, suitable for further practical applications. The use of other tolerance rules is also discussed. Springer-Verlag Berlin Heidelberg, 2000 variational inequalities - proximal point subproblems - error bound - (enlargement of) maximal monotone operator - regularized gap function Mathematics Subject Classification (1991) nationallicence 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,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,benar}@impa.br, BR aut Mathematical Programming Springer-Verlag 88/2(2000-08-01), 371-389 0025-5610 88:2<371 2000 88 10107 https://doi.org/10.1007/s101070050022 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s101070050022 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,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,benar}@impa.br, BR aut NATIONALLICENCE 773 0- Mathematical Programming Springer-Verlag 88/2(2000-08-01), 371-389 0025-5610 88:2<371 2000 88 10107 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer