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.
All Science Journal Classification (ASJC) codes
- Control and Optimization