TY - JOUR
T1 - Strong convergence of a proximal point algorithm with bounded error sequence
AU - Boikanyo, Oganeditse A.
AU - Moroşanu, Gheorghe
PY - 2013/2
Y1 - 2013/2
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
AN - SCOPUS:84873199588
SN - 1862-4472
VL - 7
SP - 415
EP - 420
JO - Optimization Letters
JF - Optimization Letters
IS - 2
ER -