Damped Techniques for the Limited Memory BFGS Method for Large-Scale Optimization

This paper is aimed to extend a certain damped technique, suitable for the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, to the limited memory BFGS method in the case of the large-scale unconstrained optimization. It is shown that the proposed technique maintains the global convergence property on...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2014-05, Vol.161 (2), p.688-699
Main Authors: Al-Baali, Mehiddin, Grandinetti, Lucio, Pisacane, Ornella
Format: Article
Language:English
Subjects:
Applications of Mathematics
Approximation
Calculus of Variations and Optimal Control
Optimization
Convergence
Convex analysis
Engineering
Iterative methods
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Optimization
Studies
Theory of Computation
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Summary:This paper is aimed to extend a certain damped technique, suitable for the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, to the limited memory BFGS method in the case of the large-scale unconstrained optimization. It is shown that the proposed technique maintains the global convergence property on uniformly convex functions for the limited memory BFGS method. Some numerical results are described to illustrate the important role of the damped technique. Since this technique enforces safely the positive definiteness property of the BFGS update for any value of the steplength, we also consider only the first Wolfe–Powell condition on the steplength. Then, as for the backtracking framework, only one gradient evaluation is performed on each iteration. It is reported that the proposed damped methods work much better than the limited memory BFGS method in several cases.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-013-0448-8