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In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient (HPE) framework was used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth variational inequalities and inclusion problems consisting of the sum of a smooth monotone map and a maximal monotone point-to-set operator. Each iteration of the method computes an approximate solution of a proximal subproblem obtained by linearizing the smooth part of the operator in the corresponding proximal equation for the original problem, which is then used to perform an extragradient step as prescribed by the HPE framework. Both pointwise and ergodic iteration-complexity results are derived for the aforementioned first-order method using corresponding results obtained here for a subfamily of the HPE framework. [ABSTRACT FROM AUTHOR]

Finite dimensional local convergence results for self-adaptive proximal point methods and nonlinear functions with multiple minimizers are generalized and extended to a Hilbert space setting. The principle assumption is a local error bound condition which relates the growth in the function to the distance to the set of minimizers. A local convergence result is established for almost exact iterates. Less restrictive acceptance criteria for the proximal iterates are also analyzed. These criteria are expressed in terms of a subdifferential of the proximal function and either a subdifferential of the original function or an iteration difference. If the proximal regularization parameter μ(x) is sufficiently small and bounded away from zero and f is sufficiently smooth, then there is local linear convergence to the set of minimizers. For a locally convex function, a convergence result similar to that for almost exact iterates is established. For a locally convex solution set and smooth functions, it is shown that if the proximal regularization parameter has the form μ(x) = β∥f′[x]∥η, where η ϵ (0, 2), then the convergence is at least superlinear if η ϵ (0, 1) and at least quadratic if η [1, 2). [ABSTRACT FROM AUTHOR]

This paper introduces a general implicit iterative method for finding zeros of a maximal monotone operator in a Hilbert space which unifies three previously studied strategies: relaxation, inertial type extrapolation and projection step. The first two strategies are intended to speed up the convergence of the standard proximal point algorithm, while the third permits one to perform inexact proximal iterations with fixed relative error tolerance. The paper establishes the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters. [ABSTRACT FROM AUTHOR]