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On Efficiently Combining Limited Memory and Trust-Region Techniques

Oleg Burdakov (oleg.burdakov***at***liu.se)
Lujin Gong (lujin.gong***at***samsung.com)
Ya-xiang Yuan (yyx***at***lsec.cc.ac.cn)
Spartak Zikrin (spartak.zikrin***at***liu.se)

Abstract: 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.

Keywords: Unconstrained Optimization; Large-scale Problems; Limited Memory Methods; Trust Region Methods; Shape-Changing Norm; Eigenvalue Decomposition

Category 1: Nonlinear Optimization (Unconstrained Optimization )

Citation: Technical Report LiTH-MAT-R--2013/13--SE, Department of Mathematics, Linkoping University, Sweden, November/2013

Download: [PDF]

Entry Submitted: 11/25/2013
Entry Accepted: 11/25/2013
Entry Last Modified: 04/10/2015

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