Testing and Non-linear Preconditioning of the Proximal Point Method

Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the...

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Published in:Applied mathematics & optimization Vol. 82; no. 2; pp. 591 - 636
Main Author: Valkonen, Tuomo
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2020
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Control
Convergence
Fixed points (mathematics)
Goal programming
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Methods
Numerical and Computational Physics
Preconditioning
Saddle points
Simulation
Splitting
Systems Theory
Theoretical
49M29
65K10
65K15
90C30
90C47
ISSN:0095-4616, 1432-0606
Online Access:Get full text
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Abstract Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the α -averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper “Block-proximal methods with spatially adapted acceleration”. In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward–backward splitting, Douglas–Rachford splitting, Newton’s method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle–Pock method.
AbstractList Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the α -averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper “Block-proximal methods with spatially adapted acceleration”. In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward–backward splitting, Douglas–Rachford splitting, Newton’s method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle–Pock method.
Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the α-averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper “Block-proximal methods with spatially adapted acceleration”. In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward–backward splitting, Douglas–Rachford splitting, Newton’s method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle–Pock method.
Author Valkonen, Tuomo
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  surname: Valkonen
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Snippet Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and...
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SubjectTerms Calculus of Variations and Optimal Control; Optimization
Control
Convergence
Fixed points (mathematics)
Goal programming
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Methods
Numerical and Computational Physics
Preconditioning
Saddle points
Simulation
Splitting
Systems Theory
Theoretical
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Title Testing and Non-linear Preconditioning of the Proximal Point Method
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