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On exact linesearch quasi-Newton methods for minimizing a quadratic function
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.ORCID iD: 0000-0002-6252-7815
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2018 (English)In: Computational optimization and applications, ISSN 0926-6003, E-ISSN 1573-2894, Vol. 69, no 1, p. 225-241Article in journal (Refereed) Published
Abstract [en]

This paper concerns exact linesearch quasi-Newton methods for minimizing a quadratic function whose Hessian is positive definite. We show that by interpreting the method of conjugate gradients as a particular exact linesearch quasi-Newton method, necessary and sufficient conditions can be given for an exact linesearch quasi-Newton method to generate a search direction which is parallel to that of the method of conjugate gradients. We also analyze update matrices and give a complete description of the rank-one update matrices that give search direction parallel to those of the method of conjugate gradients. In particular, we characterize the family of such symmetric rank-one update matrices that preserve positive definiteness of the quasi-Newton matrix. This is in contrast to the classical symmetric-rank-one update where there is no freedom in choosing the matrix, and positive definiteness cannot be preserved. The analysis is extended to search directions that are parallel to those of the preconditioned method of conjugate gradients in a straightforward manner.

Place, publisher, year, edition, pages
Springer, 2018. Vol. 69, no 1, p. 225-241
Keywords [en]
Method of conjugate gradients, Quasi-Newton method, Unconstrained quadratic program, Exact linesearch method
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-221353DOI: 10.1007/s10589-017-9940-7ISI: 000419346400009Scopus ID: 2-s2.0-85047629692OAI: oai:DiVA.org:kth-221353DiVA, id: diva2:1174919
Funder
Swedish Research Council, 621-2014-4772
Note

QC 20180117

Available from: 2018-01-17 Created: 2018-01-17 Last updated: 2018-06-12Bibliographically approved

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Forsgren, Anders

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