scholarly journals Linear convergence of first order methods for non-strongly convex optimization

2018 ◽  
Vol 175 (1-2) ◽  
pp. 69-107 ◽  
Author(s):  
I. Necoara ◽  
Yu. Nesterov ◽  
F. Glineur
Keyword(s):  
Convex Optimization ◽  
Linear Convergence ◽  
First Order ◽  
Strongly Convex ◽  
First Order Methods ◽  
Strongly Convex Optimization

scholarly journals On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces

2021 ◽  
Author(s):  
Jakub Wiktor Both
Keyword(s):  
Convex Optimization ◽  
Banach Spaces ◽  
Optimization Problems ◽  
Linear Convergence ◽  
Rates Of Convergence ◽  
Strong Convexity ◽  
Quadratic Functional ◽  
Alternating Minimization ◽  
Strongly Convex ◽  
Strongly Convex Optimization

AbstractIn this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved for (i) and (ii), whereas sublinear convergence is showed for (iii).

scholarly journals From differential equation solvers to accelerated first-order methods for convex optimization

2021 ◽  
Author(s):  
Hao Luo ◽  
Long Chen
Keyword(s):  
Differential Equation ◽  
Lyapunov Function ◽  
Convex Optimization ◽  
Convergence Rates ◽  
Optimization Problems ◽  
Point Of View ◽  
First Order ◽  
Strongly Convex ◽  
Accelerated Gradient ◽  
First Order Methods

AbstractConvergence analysis of accelerated first-order methods for convex optimization problems are developed from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient (NAG) flow, is derived from the connection between acceleration mechanism and A-stability of ODE solvers, and the exponential decay of a tailored Lyapunov function along with the solution trajectory is proved. Numerical discretizations of NAG flow are then considered and convergence rates are established via a discrete Lyapunov function. The proposed differential equation solver approach can not only cover existing accelerated methods, such as FISTA, Güler’s proximal algorithm and Nesterov’s accelerated gradient method, but also produce new algorithms for composite convex optimization that possess accelerated convergence rates. Both the convex and the strongly convex cases are handled in a unified way in our approach.

First-order methods of smooth convex optimization with inexact oracle

2013 ◽  
Vol 146 (1-2) ◽  
pp. 37-75 ◽  
Author(s):  
Olivier Devolder ◽  
François Glineur ◽  
Yurii Nesterov
Keyword(s):  
Convex Optimization ◽  
Smooth Convex ◽  
First Order ◽  
Inexact Oracle ◽  
Smooth Convex Optimization ◽  
First Order Methods

scholarly journals First-Order Methods for Convex Optimization

2021 ◽  
pp. 100015
Author(s):  
Pavel Dvurechensky ◽  
Shimrit Shtern ◽  
Mathias Staudigl
Keyword(s):  
Convex Optimization ◽  
First Order ◽  
First Order Methods
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