Direct Search Methods over Lipschitz Manifolds∗

We extend direct search methods to optimization problems that include equality constraints given by Lipschitz functions. The equality constraints are assumed to implicitly define a Lipschitz manifold. Numerically implementing the inverse (implicit) function theorem allows us to define a new problem on the tangent spaces of the manifold. We can then use a direct search method on the tangent spaces to solve the new optimization problem without any equality constraints. Solving this related problem implicitly solves the original optimization problem. Our main example utilizes the LTMADS algorithm for the direct search method. However, other direct search methods can be employed. Convergence results trivially carry over to our new procedure under mild assumptions.