On over-relaxed ( A , η , m ) -proximal point algorithm frameworks with errors and applications to general variational inclusion problems

Heng-You Lan

doi:10.1186/1029-242x-2013-97

Heng-You Lan

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https://doi.org/10.1186/1029-242x-2013-97

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The purpose of this paper is to provide some remarks for the main results of the paper Verma (Appl. Math. Lett. 21:142-147, 2008). Further, by using the generalized proximal operator technique associated with the -monotone operators, we discuss the approximation solvability of general variational inclusion problem forms in Hilbert spaces and the convergence analysis of iterative sequences generated by the over-relaxed -proximal point algorithm frameworks with errors, which generalize the hybrid proximal point algorithm frameworks due to Verma. MSC:47H05, 49J40.

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In, Verma [ ] developed a general framework for a hybrid proximal point algorithm using the notion of (A, η)-monotonicity ( referred to as (A, η)-maximal monotonicity or (A, η, m)-monotonicity in literature) and explored convergence analysis for this algorithm in the context of solving the following variational inclusion problems along with some results on the resolvent operator corresponding to (A, η)-monotonicity: Find a solution to ∈ M(x), ( . )where M : H → H is a set-valued mapping on a real Hilbert space H
Example . [ ] Let V : Rn → R be a local Lipschitz continuous function, and let K be a closed convex set in Rn
Verma [ ] pointed out ‘the over-relaxed proximal point algorithm is of interest in the sense that it is quite application-oriented, but nontrivial in nature’

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In , Verma [ ] developed a general framework for a hybrid proximal point algorithm using the notion of (A, η)-monotonicity ( referred to as (A, η)-maximal monotonicity or (A, η, m)-monotonicity in literature) and explored convergence analysis for this algorithm in the context of solving the following variational inclusion problems along with some results on the resolvent operator corresponding to (A, η)-monotonicity: Find a solution to ∈ M(x), ( . )where M : H → H is a set-valued mapping on a real Hilbert space H. In [ , ], we discussed the convergence of iterative sequences generated by the hybrid proximal point algorithm frameworks associated with (A, η, m)-monotonicity when operator A is strongly monotone and Lipschitz continuous.

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