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An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator T on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation x ˙ (t) + T λ (t) (x (t) − α (t) T λ (t) (x (t))) = 0 , t ≥ t 0 ≥ 0 , to the solution set T − 1 (0) of a generalized equation 0 ∈ T (x). With smart choices of parameters λ (t) and α (t) , we prove the weak convergence of the trajectory to some point of T − 1 (0) with ‖ x ˙ (t) ‖ ≤ O (1 / t) as t → + ∞. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory x h (t) → x (t) on interval [ 0 , + ∞) is demonstrated when the step size h → 0. [ABSTRACT FROM AUTHOR]

This paper presents an accelerated variant of the hybrid proximal extragradient (HPE) method for convex optimization, referred to as the accelerated HPE (A-HPE) framework. Iteration-complexity results are established for the A-HPE framework, as well as a special version of it, where a large stepsize condition is imposed. Two specific implementations of the A-HPE framework are described in the context of a structured convex optimization problem whose objective function consists of the sum of a smooth convex function and an extended real-valued nonsmooth convex function. In the first implementation, a generalization of a variant of Nesterov's method is obtained for the case where the smooth component of the objective function has Lipschitz continuous gradient. In the second implementation, an accelerated Newton proximal extragradient (A-NPE) method is obtained for the case where the smooth component of the objective function has Lipschitz continuous Hessian. It is shown that the A-NPE method has a O(1/k7/2) convergence rate, which improves upon the O(1/k³) convergence rate bound for another accelerated Newton-type method presented by Nesterov. Finally, while Nesterov's method is based on exact solutions of subproblems with cubic regularization terms, the A-NPE method is based on inexact solutions of subproblems with quadratic regularization terms and hence is potentially more tractable from a computational point of view. [ABSTRACT FROM AUTHOR]

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]

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]