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On Efficiently Combining Limited-Memory and Trust-Region Techniques
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .ORCID iD: 0000-0003-1836-4200
Tencent, Beijing, China.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, CAS, Beijing, China.
2016 (English)In: Mathematical Programming Computation, ISSN 1867-2949, E-ISSN 1867-2957, 1-34 p.Article in journal (Refereed) Epub ahead of print
Abstract [en]

Limited-memory quasi-Newton methods and trust-region methods represent two efficient approaches used for solving unconstrained optimization problems. A straightforward combination of them deteriorates the efficiency of the former approach, especially in the case of large-scale problems. For this reason, the limited-memory methods are usually combined with a line search. We show how to efficiently combine limited-memory and trust-region techniques. One of our approaches is based on the eigenvalue decomposition of the limited-memory quasi-Newton approximation of the Hessian matrix. The decomposition allows for finding a nearly-exact solution to the trust-region subproblem defined by the Euclidean norm with an insignificant computational overhead as compared with the cost of computing the quasi-Newton direction in line-search limited-memory methods. The other approach is based on two new eigenvalue-based norms. The advantage of the new norms is that the trust-region subproblem is separable and each of the smaller subproblems is easy to solve. We show that our eigenvalue-based limited-memory trust-region methods are globally convergent. Moreover, we propose improved versions of the existing limited-memory trust-region algorithms. The presented results of numerical experiments demonstrate the efficiency of our approach which is competitive with line-search versions of the L-BFGS method.

Place, publisher, year, edition, pages
2016. 1-34 p.
Keyword [en]
Unconstrained Optimization, Large-scale Problems, Limited-Memory Methods, Trust-Region Methods, Shape-Changing Norm, Eigenvalue Decomposition
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-129783DOI: 10.1007/s12532-016-0109-7OAI: oai:DiVA.org:liu-129783DiVA: diva2:943690
Available from: 2016-06-28 Created: 2016-06-28 Last updated: 2016-07-06Bibliographically approved

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The full text will be freely available from 2017-06-26 09:38
Available from 2017-06-26 09:38

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