The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai-Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage of the availability of a few additional 'long' vectors of storage to achieve a significant improvement in performance, both for quadratic and non-quadratic objective functions. It makes use of certain Ritz values related to the Lanczos process. The talk refers briefly to some underlying theory, and numerical evidence is described showing that the new method provides a competitive and more simple alternative to the state of the art l-BFGS limited memory method. This work is documented in ERGO report 09-014. Recent work in extending the ideas to QP and general linear constraint minimization will be described.
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