Strong convergence of a proximal-type algorithm for an occasionally pseudomonotone operator in Banach spaces

Pathak Hemant, Cho, Cho Yeol

Description

It is known that the proximal point algorithm converges weakly to a zero of a maximal monotone operator, but it fails to converge strongly. Then, in (Math. Program. 87:189-202, 2000), Solodov and Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for the algorithm in Hilbert spaces. Further, Kamimura and Takahashi (SIAM J. Optim. 13:938-945, 2003) extended Solodov and Svaiter’s result to more general Banach spaces and obtained strong convergence of a proximal-type algorithm in Banach spaces. In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi. MSC: 47H05, 47J25.

Sujets

• Monotonic function
• Banach space
• Strong convergence

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 197
• Langue: English

Extrait

Pathak and ChoFixed Point Theory and Applications2012,2012:190 http://www.fixedpointtheoryandapplications.com/content/2012/1/190

R E S E A R C HOpen Access Strong convergence of a proximal-type algorithm for an occasionally pseudomonotone operator in Banach spaces 1 2* Hemant Kumar Pathakand Yeol Je Cho

* Correspondence: [e-mail] 2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract It is known that the proximal point algorithm converges weakly to a zero of a maximal monotone operator, but it fails to converge strongly. Then, in (Math. Program. 87:189-202, 2000), Solodov and Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for the algorithm in Hilbert spaces. Further, Kamimura and Takahashi (SIAM J. Optim. 13:938-945, 2003) extended Solodov and Svaiter’s result to more general Banach spaces and obtained strong convergence of a proximal-type algorithm in Banach spaces. In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi. MSC:47H05; 47J25 Keywords:proximal point algorithm; monotone operator; maximal monotone operator; pseudomonotone operator; occasionally pseudomonotone operator; maximal pseudomonotone operator; maximal occasionally pseudomonotone operator; Banach space; strong convergence

1 Introduction H LetHbe a real Hilbert space with inner product∙,∙, and letT:H→a maximal be monotone operator (or a multifunction) onH. We consider the classical problem: Findx∈Hsuch that

∈Tx.

(.)

A wide variety of the problems, such as optimization problems and related fields, min- max problems, complementarity problems, variational inequalities, equilibrium problems and fixed point problems, fall within this general framework. For example, ifTis the subd- ifferential∂fof a proper lower semicontinuous convex functionf:H→(–∞,∞), thenT is a maximal monotone operator and the equation ∈∂f(x) is reduced tof(x) =min{f(z) : z∈H}. One method of solving ∈Txis the proximal point algorithm. LetIdenote the identity operator onH. Rockafellar’s proximal point algorithm generates, for any starting pointx  =x∈H, a sequence{x n }inHby the rule

– x n+ = (I+r n T)x n ,∀n≥, (.) ©2012 Pathak and Cho; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.