Strong convergence of a proximal point algorithm with bounded error sequence

Oganeditse A. Boikanyo, Gheorghe Moroşanu

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Given any maximal monotone operator A: D(A) ⊂ H → 2H in a real Hilbert space H with A-1(0) ≠ ∅, it is shown that the sequence of proximal iterates xn+1 = (I+ γnA)-1nu + (1-λn)(xn+en)) converges strongly to the metric projection of u on A-1(0) for (en) bounded, λn ∈ (0,1) with λn → 1 and γn > 0 with γn → ∞ as n → ∞. In comparison with our previous paper (Boikanyo and Moroşanu in Optim Lett 4(4):635-641, 2010), where the error sequence was supposed to converge to zero, here we consider the classical condition that errors be bounded. In the case when A is the subdifferential of a proper convex lower semicontinuous function φ: H → (-∞,+∞] the algorithm can be used to approximate the minimizer of φ which is nearest to u.

Original languageEnglish
Pages (from-to)415-420
Number of pages6
JournalOptimization Letters
Volume7
Issue number2
DOIs
Publication statusPublished - Jan 1 2013

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Proximal Point Algorithm
Strong Convergence
Converge
Metric Projection
Lower Semicontinuous Function
Maximal Monotone Operator
Subdifferential
Minimizer
Iterate
Convex function
Hilbert space
Zero

All Science Journal Classification (ASJC) codes

  • Control and Optimization

Cite this

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Strong convergence of a proximal point algorithm with bounded error sequence. / Boikanyo, Oganeditse A.; Moroşanu, Gheorghe.

In: Optimization Letters, Vol. 7, No. 2, 01.01.2013, p. 415-420.

Research output: Contribution to journalArticle

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