Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.

The adaptive cubic regularization algorithms described in Cartis, Gould and Toint [Adaptive cubic regularisation methods for unconstrained optimization Part II: Worst-case function- and derivative-evaluation complexity, Math. Program. (2010), doi:10.1007/s10107-009-0337-y (online)]; [Part I: Motivat...

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Main Authors: Cartis, C, Gould, N, Toint, P
Format: Journal article
Language:English
Published: 2012
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author Cartis, C
Gould, N
Toint, P
author_facet Cartis, C
Gould, N
Toint, P
author_sort Cartis, C
collection OXFORD
description The adaptive cubic regularization algorithms described in Cartis, Gould and Toint [Adaptive cubic regularisation methods for unconstrained optimization Part II: Worst-case function- and derivative-evaluation complexity, Math. Program. (2010), doi:10.1007/s10107-009-0337-y (online)]; [Part I: Motivation, convergence and numerical results, Math. Program. 127(2) (2011), pp. 245-295] for unconstrained (nonconvex) optimization are shown to have improved worst-case efficiency in terms of the function- and gradient-evaluation count when applied to convex and strongly convex objectives. In particular, our complexity upper bounds match in order (as a function of the accuracy of approximation), and sometimes even improve, those obtained by Nesterov [Introductory Lectures on Convex Optimization, Kluwer Academic Publishers, Dordrecht, 2004; Accelerating the cubic regularization of Newton's method on convex problems, Math. Program. 112(1) (2008), pp. 159-181] and Nesterov and Polyak [Cubic regularization of Newton's method and its global performance, Math. Program. 108(1) (2006), pp. 177-205] for these same problem classes, without requiring exact Hessians or exact or global solution of the subproblem. An additional outcome of our approximate approach is that our complexity results can naturally capture the advantages of both first- and second-order methods. © 2012 Taylor and Francis.
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spelling oxford-uuid:8dc00e5a-249f-44f5-815c-dd39e38f2d5b2022-03-26T22:53:09ZEvaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:8dc00e5a-249f-44f5-815c-dd39e38f2d5bEnglishSymplectic Elements at Oxford2012Cartis, CGould, NToint, PThe adaptive cubic regularization algorithms described in Cartis, Gould and Toint [Adaptive cubic regularisation methods for unconstrained optimization Part II: Worst-case function- and derivative-evaluation complexity, Math. Program. (2010), doi:10.1007/s10107-009-0337-y (online)]; [Part I: Motivation, convergence and numerical results, Math. Program. 127(2) (2011), pp. 245-295] for unconstrained (nonconvex) optimization are shown to have improved worst-case efficiency in terms of the function- and gradient-evaluation count when applied to convex and strongly convex objectives. In particular, our complexity upper bounds match in order (as a function of the accuracy of approximation), and sometimes even improve, those obtained by Nesterov [Introductory Lectures on Convex Optimization, Kluwer Academic Publishers, Dordrecht, 2004; Accelerating the cubic regularization of Newton's method on convex problems, Math. Program. 112(1) (2008), pp. 159-181] and Nesterov and Polyak [Cubic regularization of Newton's method and its global performance, Math. Program. 108(1) (2006), pp. 177-205] for these same problem classes, without requiring exact Hessians or exact or global solution of the subproblem. An additional outcome of our approximate approach is that our complexity results can naturally capture the advantages of both first- and second-order methods. © 2012 Taylor and Francis.
spellingShingle Cartis, C
Gould, N
Toint, P
Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.
title Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.
title_full Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.
title_fullStr Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.
title_full_unstemmed Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.
title_short Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.
title_sort evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization
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