Summary: A review is given of the underlying theory and recent developments in regard to the Barzilai-Borwein steepest descent method for large scale unconstrained optimization. One aim is to assess why the method seems to be comparable in practical efficiency to conjugate gradient methods. The importance of using a non-monotone line search is stressed, although some suggestions are made as to why the modification proposed by Raydan (1997) often does not usually perform well for an ill-conditioned problem. Extensions for box constraints are discussed. A number of interesting open questions are put forward.
For the entire collection see [Zbl 1060.91002].