Strong convergence of a proximal point algorithm with bounded error sequence

Abstract

Given any maximal monotone operator ${A: D(A)\subset H \rightarrow 2^H}$ in a real Hilbert space H with ${A^{-1}(0) \ne \emptyset}$ , it is shown that the sequence of proximal iterates ${x_{n+1}=(I+\gamma_n A)^{-1}(\lambda_n u+(1-\lambda_n)(x_n+e_n))}$ converges strongly to the metric projection of u on A ⁻¹(0) for (e _n ) bounded, ${\lambda_n \in (0,1)}$ with ${\lambda_n \to 1}$ and γ _n  > 0 with ${\gamma_n \to\infty}$ as ${n \to \infty}$ . 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 ${\varphi :H \to (-\infty,+ \infty]}$ , the algorithm can be used to approximate the minimizer of φ which is nearest to u.