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Efficient first-order methods for convex minimization: a constructive approach

We describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a certain variant of the conjugate gradient method to construc... Full description

Journal Title: Mathematical Programming Series A, 2019, Vol.184 (1-2), p.183-220
Main Author: Drori, Yoel
Other Authors: Taylor, Adrien B
Format: Electronic Article Electronic Article
Language: English
Subjects:
Publisher: Berlin/Heidelberg: Springer Berlin Heidelberg
ID: ISSN: 0025-5610
Link: https://hal.inria.fr/hal-01902048
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recordid: cdi_proquest_journals_2245977358
title: Efficient first-order methods for convex minimization: a constructive approach
format: Article
creator:
  • Drori, Yoel
  • Taylor, Adrien B
subjects:
  • Calculus of Variations and Optimal Control
  • Optimization
  • Combinatorics
  • Computer Science
  • Conjugate gradient method
  • Conjugates
  • Construction methods
  • Control
  • Full Length Paper
  • Iterative methods
  • Mathematical and Computational Physics
  • Mathematical Methods in Physics
  • Mathematics
  • Mathematics and Statistics
  • Mathematics of Computing
  • Methods
  • Numerical Analysis
  • Optimization
  • Optimization and Control
  • Theoretical
ispartof: Mathematical Programming, Series A, 2019, Vol.184 (1-2), p.183-220
description: We describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a certain variant of the conjugate gradient method to construct a family of methods such that (a) all methods in the family share the same worst-case guarantee as the base conjugate gradient method, and (b) the family includes a fixed-step first-order method. We demonstrate the effectiveness of the approach by deriving optimal methods for the smooth and non-smooth cases, including new methods that forego knowledge of the problem parameters at the cost of a one-dimensional line search per iteration, and a universal method for the union of these classes that requires a three-dimensional search per iteration. In the strongly convex case, we show how numerical tools can be used to perform the construction, and show that the resulting method offers an improved worst-case bound compared to Nesterov’s celebrated fast gradient method.
language: eng
source:
identifier: ISSN: 0025-5610
fulltext: no_fulltext
issn:
  • 0025-5610
  • 1436-4646
  • 2331-8422
url: Link


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descriptionWe describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a certain variant of the conjugate gradient method to construct a family of methods such that (a) all methods in the family share the same worst-case guarantee as the base conjugate gradient method, and (b) the family includes a fixed-step first-order method. We demonstrate the effectiveness of the approach by deriving optimal methods for the smooth and non-smooth cases, including new methods that forego knowledge of the problem parameters at the cost of a one-dimensional line search per iteration, and a universal method for the union of these classes that requires a three-dimensional search per iteration. In the strongly convex case, we show how numerical tools can be used to perform the construction, and show that the resulting method offers an improved worst-case bound compared to Nesterov’s celebrated fast gradient method.
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subjectCalculus of Variations and Optimal Control; Optimization ; Combinatorics ; Computer Science ; Conjugate gradient method ; Conjugates ; Construction methods ; Control ; Full Length Paper ; Iterative methods ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Methods ; Numerical Analysis ; Optimization ; Optimization and Control ; Theoretical
ispartofMathematical Programming, Series A, 2019, Vol.184 (1-2), p.183-220
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2Mathematical Programming is a copyright of Springer, (2019). All Rights Reserved.
3Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019.
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descriptionWe describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a certain variant of the conjugate gradient method to construct a family of methods such that (a) all methods in the family share the same worst-case guarantee as the base conjugate gradient method, and (b) the family includes a fixed-step first-order method. We demonstrate the effectiveness of the approach by deriving optimal methods for the smooth and non-smooth cases, including new methods that forego knowledge of the problem parameters at the cost of a one-dimensional line search per iteration, and a universal method for the union of these classes that requires a three-dimensional search per iteration. In the strongly convex case, we show how numerical tools can be used to perform the construction, and show that the resulting method offers an improved worst-case bound compared to Nesterov’s celebrated fast gradient method.
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abstractWe describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a certain variant of the conjugate gradient method to construct a family of methods such that (a) all methods in the family share the same worst-case guarantee as the base conjugate gradient method, and (b) the family includes a fixed-step first-order method. We demonstrate the effectiveness of the approach by deriving optimal methods for the smooth and non-smooth cases, including new methods that forego knowledge of the problem parameters at the cost of a one-dimensional line search per iteration, and a universal method for the union of these classes that requires a three-dimensional search per iteration. In the strongly convex case, we show how numerical tools can be used to perform the construction, and show that the resulting method offers an improved worst-case bound compared to Nesterov’s celebrated fast gradient method.
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doi10.1007/s10107-019-01410-2
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