A modified Hestenes–Stiefel projection method for constrained nonlinear equations and its linear convergence rate

2014 ◽  
Vol 49 (1-2) ◽  
pp. 145-156 ◽  
Author(s):  
Min Sun ◽  
Jing Liu
Keyword(s):  
Convergence Rate ◽  
Nonlinear Equations ◽  
Projection Method ◽  
Linear Convergence ◽  
Linear Convergence Rate ◽  
Constrained Nonlinear Equations

scholarly journals Convergence rate analysis of an iterative algorithm for solving the multiple-sets split equality problem

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shijie Sun ◽  
Meiling Feng ◽  
Luoyi Shi
Keyword(s):  
Convergence Rate ◽  
Iterative Algorithm ◽  
Sufficient Conditions ◽  
Linear Convergence ◽  
Regularity Property ◽  
Step Size ◽  
Bounded Linear ◽  
Split Equality Problem ◽  
Linear Convergence Rate ◽  
Equality Problem

Abstract This paper considers an iterative algorithm of solving the multiple-sets split equality problem (MSSEP) whose step size is independent of the norm of the related operators, and investigates its sublinear and linear convergence rate. In particular, we present a notion of bounded Hölder regularity property for the MSSEP, which is a generalization of the well-known concept of bounded linear regularity property, and give several sufficient conditions to ensure it. Then we use this property to conclude the sublinear and linear convergence rate of the algorithm. In the end, some numerical experiments are provided to verify the validity of our consequences.

A modified limited memory steepest descent method motivated by an inexact super-linear convergence rate analysis

2020 ◽  
Author(s):  
Ran Gu ◽  
Qiang Du
Keyword(s):  
Convergence Rate ◽  
Linear Convergence ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Step Size ◽  
Linear Convergence Rate ◽  
Limited Memory ◽  
Rate Analysis ◽  
Convergence Rate Analysis

Abstract How to choose the step size of gradient descent method has been a popular subject of research. In this paper we propose a modified limited memory steepest descent method (MLMSD). In each iteration we propose a selection rule to pick a unique step size from a candidate set, which is calculated by Fletcher’s limited memory steepest descent method (LMSD), instead of going through all the step sizes in a sweep, as in Fletcher’s original LMSD algorithm. MLMSD is motivated by an inexact super-linear convergence rate analysis. The R-linear convergence of MLMSD is proved for a strictly convex quadratic minimization problem. Numerical tests are presented to show that our algorithm is efficient and robust.

scholarly journals Surpassing Gradient Descent Provably: A Cyclic Incremental Method with Linear Convergence Rate

2018 ◽  
Vol 28 (2) ◽  
pp. 1420-1447 ◽  
Author(s):  
Aryan Mokhtari ◽  
Mert Gürbüzbalaban ◽  
Alejandro Ribeiro
Keyword(s):  
Convergence Rate ◽  
Gradient Descent ◽  
Linear Convergence ◽  
Linear Convergence Rate ◽  
Incremental Method

scholarly journals A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Min Sun ◽  
Jing Liu
Keyword(s):  
Convergence Rate ◽  
Large Scale ◽  
Gradient Methods ◽  
Linear Convergence ◽  
New Method ◽  
Linear Convergence Rate ◽  
Relax Factor ◽  
Constrained Equations ◽  
Accelerate Convergence ◽  
Matrix Free

A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak) conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factorγis attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.

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