Sizing the BFGS and DFP Updates: A Numerical Study
In this study we develop and test a strategy for selectively sizing (multiplying by an appropriate scalar) the approximate Hessian matrix before it is updated in the BFGS and DFP trust-region methods for unconstrained optimization. Our numerical results imply that for use with the DFP update the Oren-Luenberger sizing factor is completely satisfactory and selective sizing is vastly superior to the alternatives of never sizing, or first-iteration sizing, and is slightly better than the alternative of always sizing. Numerical experimentation showed that the Oren-Luenberger sizing factor is not a satisfactory sizing factor for use with the BFGS update. Therefore, based on our newly acquired understanding of the situation, we propose a damped Oren-Luenberger sizing factor to be used with the BFGS update. Our numerical experimentation implies that selectively sizing the BFGS update with the damped Oren-Luenberger sizing factor is superior to the alternatives. These results contradict the folk-axiom that sizing should be done only at the first iteration. They also show that without sufficient sizing, DFP is vastly inferior to BFGS; however, when selectively sized, DFP is competitive with BFGS.
Citable link to this pagehttps://hdl.handle.net/1911/101721
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- CAAM Technical Reports