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Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity

An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi: 10.1007/s10107-009-0286-5 , 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981... Full description

Journal Title: Mathematical programming 2010, Vol.130 (2), p.295-319
Main Author: Cartis, Coralia
Other Authors: Gould, Nicholas I. M , Toint, Philippe L
Format: Electronic Article Electronic Article
Language: English
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Publisher: Berlin/Heidelberg: Springer-Verlag
ID: ISSN: 0025-5610
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title: Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity
format: Article
creator:
  • Cartis, Coralia
  • Gould, Nicholas I. M
  • Toint, Philippe L
subjects:
  • Algorithms
  • Analysis
  • Applied sciences
  • Approximation
  • Calculus of variations and optimal control
  • Calculus of Variations and Optimal Control
  • Optimization
  • Combinatorics
  • Complexity
  • Curvature
  • Exact sciences and technology
  • Full Length Paper
  • Iterative methods
  • Laboratories
  • Mathematical analysis
  • Mathematical and Computational Physics
  • Mathematical Methods in Physics
  • Mathematical models
  • Mathematical optimization
  • Mathematical programming
  • Mathematics
  • Mathematics and Statistics
  • Mathematics of Computing
  • Methods
  • Norms
  • Numerical Analysis
  • Operational research and scientific management
  • Operational research. Management science
  • Optimization
  • Sciences and techniques of general use
  • Studies
  • Theoretical
ispartof: Mathematical programming, 2010, Vol.130 (2), p.295-319
description: An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi: 10.1007/s10107-009-0286-5 , 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177–205, 2006) and a proposal by Weiser, Deuflhard and Erdmann (Optim Methods Softw 22(3):413–431, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy , and iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm 3.3 of Nesterov and Polyak which minimizes the cubic model globally on each iteration. Our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate Hessians.
language: eng
source:
identifier: ISSN: 0025-5610
fulltext: no_fulltext
issn:
  • 0025-5610
  • 1436-4646
url: Link


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titleAdaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity
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descriptionAn Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi: 10.1007/s10107-009-0286-5 , 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177–205, 2006) and a proposal by Weiser, Deuflhard and Erdmann (Optim Methods Softw 22(3):413–431, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy , and iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm 3.3 of Nesterov and Polyak which minimizes the cubic model globally on each iteration. Our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate Hessians.
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descriptionAn Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi: 10.1007/s10107-009-0286-5 , 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177–205, 2006) and a proposal by Weiser, Deuflhard and Erdmann (Optim Methods Softw 22(3):413–431, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy , and iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm 3.3 of Nesterov and Polyak which minimizes the cubic model globally on each iteration. Our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate Hessians.
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abstractAn Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi: 10.1007/s10107-009-0286-5 , 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177–205, 2006) and a proposal by Weiser, Deuflhard and Erdmann (Optim Methods Softw 22(3):413–431, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy , and iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm 3.3 of Nesterov and Polyak which minimizes the cubic model globally on each iteration. Our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate Hessians.
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pubSpringer-Verlag
doi10.1007/s10107-009-0337-y