A new approximate proximal point algorithm for maximal monotone operator

The problem concerned in this paper is the set-valued equation 0 is an element of T(z) where T is a maximal monotone operator. For given x(k) and beta(k) > 0, some existing approximate proximal point algorithms take x(k+1) = (x) over tilde (k) such that x(k) + e(k) is an element of (x) over tilde (k) + beta(k)T((x) over tilde (k)) and parallel toe(k)parallel to less than or equal to etakparallel tox(k) - (x) over tilde (k)parallel to, where {etak} is a non-negative summable sequence. Instead of xk+1 = xk, the new iterate of the proposing method is given by x(k+1) = POmega[(x) over tilde (k) - e(k)], where Omega is the domain of T and POmega(-) denotes the projection on Omega. The convergence is proved under a significantly relaxed restriction sup(k>0) etak < 1.