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A new approximate proximal point algorithm for maximal monotone operator. (English) Zbl 1217.65102
Summary: The problem concerned in this paper is the set-valued equation 0T(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 =x ˜ k such that x k +e k x ˜ k +b k T(x ˜ k ) and e k η k x k -x ˜ k , where {η k } is a non-negative summable sequence. Instead of x k+1 =x ˜ k , the new iterate of the proposing method is given by x k+1 =P Ω [x ˜ k -e k ], where Ω is the domain of T and P Ω (·) denotes the projection on Ω. The convergence is proved under a significantly relaxed restriction supK>0ηKη1.
MSC:
65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
90C47Minimax problems