A direct search method for the class of problems considered by Lewis and Torczon [\textit{SIAM J. Optim.}, 12 (2002), pp. 1075-1089] is developed. Instead of using an augmented Lagrangian method, a simplicial approximation method to the feasible set is implicitly employed. This allows the points our algorithm considers to conveniently remain within an \textit{a priori} specified distance of the feasible set. In the limit, a positive spanning set is constructed in the tangent plane to the feasible set of every cluster point. In this way, we can guarantee that every cluster point is a stationary point for the objective function restricted to the feasible set.