Abstract and Applied Analysis
Volume 2013 (2013), Article ID 532041, 5 pages
http://dx.doi.org/10.1155/2013/532041
Research Article

Scaled Diagonal Gradient-Type Method with Extra Update for Large-Scale Unconstrained Optimization

Mahboubeh Farid,1 Wah June Leong,1 Najmeh Malekmohammadi,2 and Mustafa Mamat3

1Department of Mathematics, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran 1418765663, Iran
3Department of Mathematics, Faculty of Science and Technology, University Malaysia Terengganu, 21030 Kuala Terengganu, Malaysia

Received 18 December 2012; Revised 26 February 2013; Accepted 26 February 2013

Academic Editor: Guanglu Zhou

Copyright © 2013 Mahboubeh Farid et al. 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

We present a new gradient method that uses scaling and extra updating within the diagonal updating for solving unconstrained optimization problem. The new method is in the frame of Barzilai and Borwein (BB) method, except that the Hessian matrix is approximated by a diagonal matrix rather than the multiple of identity matrix in the BB method. The main idea is to design a new diagonal updating scheme that incorporates scaling to instantly reduce the large eigenvalues of diagonal approximation and otherwise employs extra updates to increase small eigenvalues. These approaches give us a rapid control in the eigenvalues of the updating matrix and thus improve stepwise convergence. We show that our method is globally convergent. The effectiveness of the method is evaluated by means of numerical comparison with the BB method and its variant.