On the Complexity of the Hybrid Proximal Extragradient Method for the Iterates and the Ergodic Mean

2010 ◽  
Vol 20 (6) ◽  
pp. 2755-2787 ◽  
Author(s):  
Renato D. C. Monteiro ◽  
B. F. Svaiter
Keyword(s):  
Extragradient Method ◽  
Hybrid Proximal Extragradient Method ◽  
Ergodic Mean

scholarly journals Reachability of optimal convergence rate estimates for high-order numerical convex optimization methods

2019 ◽  
Vol 484 (6) ◽  
pp. 667-671
Author(s):  
A. V. Gasnikov ◽  
E. A. Gorbunov ◽  
D. A. Kovalev ◽  
A. A. M. Mokhammed ◽  
E. A. Chernousova
Keyword(s):  
Convex Optimization ◽  
Convergence Rate ◽  
Optimization Problems ◽  
Extragradient Method ◽  
Auxiliary Problem ◽  
Optimization Methods ◽  
Step Size ◽  
Tensor Method ◽  
Higher Derivatives ◽  
Hybrid Proximal Extragradient Method

The Monteiro-Svaiter accelerated hybrid proximal extragradient method (2013) with one step of Newton’s method used at every iteration for the approximate solution of an auxiliary problem is considered. The Monteiro-Svaiter method is optimal (with respect to the number of gradient and Hessian evaluations for the optimized function) for sufficiently smooth convex optimization problems in the class of methods using only the gradient and Hessian of the optimized function. An optimal tensor method involving higher derivatives is proposed by replacing Newton’s step with a step of Yu.E. Nesterov’s recently proposed tensor method (2018) and by using a special generalization of the step size selection condition in the outer accelerated proximal extragradient method. This tensor method with derivatives up to the third order inclusive is found fairly practical, since the complexity of its iteration is comparable with that of Newton’s one. Thus, a constructive solution is obtained for Nesterov’s problem (2018) of closing the gap between tight lower and overstated upper bounds for the convergence rate of existing tensor methods of order p ≥ 3.

scholarly journals A Mean Extragradient Method for Solving Variational Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 462
Author(s):  
Apichit Buakird ◽  
Nimit Nimana ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Mean Value ◽  
Inequality Problem ◽  
The Mean ◽  
Mean Value Method

We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.

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