9.3.4 Direct

The DIRECT algorithm is an algorithm for finding the extrema of a Lipschitz continuous function (cf. Definition A.8) in a multidimensional interval [58,37]. The existence of derivatives is not required.

Classic Lipschitzian optimization is based on so called saw-tooth covers of the given function $f$, which can be deduced from its Lipschitz constant and evaluations of $f$ by dividing intervals. The DIRECT algorithm, introduced in [58], differs from classic Lipschitzian optimization [91,110] in its choice of new points when dividing intervals and in estimating the Lipschitz constant which is not necessarily known.

The DIRECT algorithm exploits the search space well and has a good global behavior, but a bad local behavior [37]. Because of this reason and evidence from experiments performed, it is a good starting point generator in many cases. An interface to the implementation described in [37] is provided in SIESTA.