Asymptotic convergence analysis of the proximal point algorithm for metrically regular mappings

This paper studies convergence properties of the proximal point algorithm when applied to a certain class of nonmonotone set-valued mappings. We consider an algorithm for solving an inclusion 0 ∈ T(x), where T is a metrically regular set-valued mapping acting from R(n) into R(m). The algorithm is given by the follwoing iteration: x(0) ∈ R(n) and x(k+1) = α(k)x(k) + (1 - α(k))y(k), for k = 0, 1, 2, ..., where {α(k)} is a sequence in [0, 1] such that α(k) ≤ α < 1, g(k) is a Lipschitz mapping from R(n) into R(m) and y(k) satisfies the following inclusion 0 ∈ g(k)(y(k)) - g(k)(x(k)) + T(y(k)). We prove that if the modulus of regularity of T is sufficiently small then the sequence generated by our algorithm converges to a solution to 0 ∈ T(x).