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)

Hybrid Proximal Methods for Equilibrium Problems

This paper concerns developing two hybrid proximal point methods (PPMs) for finding a common solution of some optimization-related problems. First we construct an algorithm to solve simultaneously an equilibrium problem and a variational inequality problem, combing the extragradient method for variational inequalities with an approximate PPM for equilibrium problems. Next we develop another algorithm based on an alternate approximate PPM for finding a common solution of two different equilibrium problems. We prove the global convergence of both algorithms under pseudomonotonicity assumptions

A proximal method for composite minimization

Abstract. We consider minimization of functions that are compositions of prox-regular functions with smooth vector functions. A wide variety of important optimization problems can be formulated in this way. We describe a subproblem constructed from a linearized approximation to the objective and a regularization term, investigating the properties of local solutions of this subproblem and showing that they eventually identify a manifold containing the solution of the original problem. We propose an algorithmic framework based on this subproblem and prove a global convergence result