A variation of Broyden class methods using Householder adaptive transforms
A variation of Broyden class methods using Householder adaptive transforms
In this work we introduce and study novel Quasi Newton minimization methods based on a Hessian approximation Broyden Class-type updating scheme, where a suitable matrix B~ k is updated instead of the current Hessian approximation Bk. We identify conditions which imply the convergence of the algorithm and, if exact line search is chosen, its quadratic termination. By a remarkable connection between the projection operation and Krylov spaces, such conditions can be ensured using low complexity matrices B~ k obtained projecting Bk onto algebras of matrices diagonalized by products of two or three Householder matrices adaptively chosen step by step. Experimental tests show that the introduction of the adaptive criterion, which theoretically guarantees the convergence, considerably improves the robustness of the minimization schemes when compared with a non-adaptive choice; moreover, they show that the proposed methods could be particularly suitable to solve large scale problems where L-BFGS is not able to deliver satisfactory performance.
Matrix algebras, Matrix projections preserving directions, Quasi-Newton methods, Unconstrained minimization
433-463
Cipolla, S.
373fdd4b-520f-485c-b36d-f75ce33d4e05
Di Fiore, C.
f43c7f86-7a2e-474a-ad41-3ff3797128bc
Zellini, P.
ba2b701f-50cd-4b28-a91f-5bcd86484960
1 November 2020
Cipolla, S.
373fdd4b-520f-485c-b36d-f75ce33d4e05
Di Fiore, C.
f43c7f86-7a2e-474a-ad41-3ff3797128bc
Zellini, P.
ba2b701f-50cd-4b28-a91f-5bcd86484960
Cipolla, S., Di Fiore, C. and Zellini, P.
(2020)
A variation of Broyden class methods using Householder adaptive transforms.
Computational Optimization and Applications, 77 (2), .
(doi:10.1007/s10589-020-00209-8).
Abstract
In this work we introduce and study novel Quasi Newton minimization methods based on a Hessian approximation Broyden Class-type updating scheme, where a suitable matrix B~ k is updated instead of the current Hessian approximation Bk. We identify conditions which imply the convergence of the algorithm and, if exact line search is chosen, its quadratic termination. By a remarkable connection between the projection operation and Krylov spaces, such conditions can be ensured using low complexity matrices B~ k obtained projecting Bk onto algebras of matrices diagonalized by products of two or three Householder matrices adaptively chosen step by step. Experimental tests show that the introduction of the adaptive criterion, which theoretically guarantees the convergence, considerably improves the robustness of the minimization schemes when compared with a non-adaptive choice; moreover, they show that the proposed methods could be particularly suitable to solve large scale problems where L-BFGS is not able to deliver satisfactory performance.
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Accepted/In Press date: 14 July 2020
Published date: 1 November 2020
Additional Information:
Funding Information: the authors acknowledge anonymous referees for their thorough reading of the manuscript and the many suggestions they gave in order to improve its readability. Moreover, they acknowledge the Associated Editor for his/her valuable commentaries and for suggesting the introduction of the scaling factor as in Sect. . S.C. and C.D.F. are members of the INdAM Research group GNCS, which partially supported this work. C.D.F acknowledges the partial support of the Italian mathematics Research Institute INdAM-GNCS and of the MIUR Excellence Department Project awarded to the Dept of Mathematics, Univ. of Rome “Tor Vergata”, CUP E83C18000100006.
Keywords:
Matrix algebras, Matrix projections preserving directions, Quasi-Newton methods, Unconstrained minimization
Identifiers
Local EPrints ID: 485532
URI: http://eprints.soton.ac.uk/id/eprint/485532
ISSN: 0926-6003
PURE UUID: e1baa1e3-dc4c-488b-b237-a32aa6f8fea5
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Date deposited: 08 Dec 2023 17:41
Last modified: 18 Mar 2024 04:17
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Author:
S. Cipolla
Author:
C. Di Fiore
Author:
P. Zellini
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