### Abstract

Given any maximal monotone operator A: D(A) ⊂ H → 2^{H} in a real Hilbert space H with A^{-1}(0) ≠ ∅, it is shown that the sequence of proximal iterates x_{n+1} = (I+ γ_{n}A)^{-1}(λ_{n}u + (1-λ_{n})(x_{n}+e_{n})) converges strongly to the metric projection of u on A^{-1}(0) for (e_{n}) bounded, λ_{n} ∈ (0,1) with λ_{n} → 1 and γ_{n} > 0 with γ_{n} → ∞ 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 language | English |
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Pages (from-to) | 415-420 |

Number of pages | 6 |

Journal | Optimization Letters |

Volume | 7 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Control and Optimization

### Cite this

*Optimization Letters*,

*7*(2), 415-420. https://doi.org/10.1007/s11590-011-0418-8

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*Optimization Letters*, vol. 7, no. 2, pp. 415-420. https://doi.org/10.1007/s11590-011-0418-8

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong convergence of a proximal point algorithm with bounded error sequence

AU - Boikanyo, Oganeditse A.

AU - Moroşanu, Gheorghe

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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)-1(λnu + (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.

AB - 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)-1(λnu + (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.

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U2 - 10.1007/s11590-011-0418-8

DO - 10.1007/s11590-011-0418-8

M3 - Article

VL - 7

SP - 415

EP - 420

JO - Optimization Letters

JF - Optimization Letters

SN - 1862-4472

IS - 2

ER -