A new approximate proximal point algorithm for maximal monotone operator

The problem concerned in this paper is the set-valued equation 0 ∈ T(z) where T is a maximal monotone operator. For given x^k and β_k >: 0, some existing approximate proximal point algorithms take \(x^{k + 1} = \tilde x^k \) such that

$x^k + e^k \in \tilde x^k + \beta _k T(\tilde x^k ) and \left\| {e^k } \right\| \leqslant \eta _k \left\| {x^k - \tilde x^k } \right\|,$

where _?k is a non-negative summable sequence. Instead of \(x^{k + 1} = \tilde x^k \), the new iterate of the proposing method is given by

$x^{k + 1} = P_\Omega \tilde x^k - e^k ],$

where Ω is the domain of T and PΩ(·) denotes the projection on Ω. The convergence is proved under a significantly relaxed restriction supK>0 ηKη1.