An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods (Q2848187)

The authors describe an accelerated variant of the hybrid proximal extragradient method in the context of convex optimization and study its iteration complexity. The proposed approach is based on the ideas from \textit{R. D. C. Monteiro} and \textit{B. F. Svaiter} [SIAM J. Optim. 20, No. 6, 2755--2787 (2010; Zbl 1230.90200)], \textit{Yu. Nesterov} [Math. Program. 103, No. 1 (A), 127--152 (2005; Zbl 1079.90102)], \textit{M. V. Solodov} and \textit{B. F. Svaiter} [Set-Valued Anal. 7, No. 4, 323--345 (1999; Zbl 0959.90038); J. Convex Anal. 6, No. 1, 59--70 (1999; Zbl 0961.90128)]. Convergence results are obtained for the presented method with a large stepsize. The authors analyse a first-order implementation of their method for solving structured convex optimization problems and obtain a generalization for some special case of Nesterov's method. They analyse also a second-order implementation of the proposed method for solving a monotone nonlinear equation. As a result, they obtain an accelerated Newton proximal extragradient method and compute its stepsize.