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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 523476, 8 pages
http://dx.doi.org/10.1155/2013/523476
Research Article

Diagonal Hessian Approximation for Limited Memory Quasi-Newton via Variational Principle

1Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2Department of Mathematics, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 23 July 2013; Revised 21 October 2013; Accepted 19 November 2013

Academic Editor: Martin Weiser

Copyright © 2013 Siti Mahani Marjugi and Wah June Leong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper proposes some diagonal matrices that approximate the (inverse) Hessian by parts using the variational principle that is analogous to the one employed in constructing quasi-Newton updates. The way we derive our approximations is inspired by the least change secant updating approach, in which we let the diagonal approximation be the sum of two diagonal matrices where the first diagonal matrix carries information of the local Hessian, while the second diagonal matrix is chosen so as to induce positive definiteness of the diagonal approximation at a whole. Some numerical results are also presented to illustrate the effectiveness of our approximating matrices when incorporated within the L-BFGS algorithm.