Tamara G. Kolda, Robert Michael Lewis, and Virginia Torczon
We show that for derivative-free, generating set search (GSS) methods for linearly constrained optimization there is a measure of stationarity that is of the same order as the step length at an identifiable subsequence of the iterates. Thus, even in the absence of explicit gradient information, we will have information about stationarity. These results help both to unify the convergence analysis of several direct search algorithms for linearly constrained problems and to clarify the fundamental geometrical ideas that underlie them. In addition, these results validate a practical stopping criterion for such algorithms, which numerical results confirm.