The authors study the so-called preconvex part of Broyden’s family, corresponding to values \(\phi \in (\phi^*,O]\) (where \(\phi^*\) is the degenerate value, \(\phi =O\) corresponds to BFGS and \(\phi =1\) to DFP update). By refining Powell’s arguments, they extend his global convergence theorem to the preconvex class and formulate a varying- parameter algorithm which at each iteration takes \(\phi_ k\) so as to obtain the steepest descent direction from a given set of quasi-Newton directions. Extensive computer experiments show that the algorithm, in the authors’ own words, “outperforms BFGS by a considerable margin on the selected test problems for the given line-search algorithm”.