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**Abstract** - A few months ago, while preparing a lecture to an audience that included engineers and umerical analysts, I asked myself the question: from the point of view of a user of onlinear optimization routines, how interesting and practical is the body of theoretical

analysis developed in this eld? To make the question a bit more precise, I decided to select the best optimization methods known to date { those methods that deserve to be n a subroutine library { and for each method ask: what do we know about the behavior of this method, as implemented in practice? To make my task more tractable, I decided o consider only algorithms for unconstrainedÂ ptimization.