Equality Constraints, Riemannian Manifolds and Direct Search Methods∗

Wepresent a general procedure for handling equality constraints in optimization problems that is of particular use in direct search methods. First we will provide the necessary background in differential geometry. In particular, we will see what a Riemannian manifold is, what a tangent space is, how to move over a manifold and how to pullback functions from a manifold to the tangent spaces. The central idea of our optimization procedure is to treat the equality constraints as implicitly defining a C Riemannian manifold. Then the function and inequality constraints can be pulled-back to the tangent spaces of this manifold. One then needs to deal with the resulting optimization problem that only involves, at most, inequality constraints. An advantage of this procedure is the implicit reduction in dimensionality of the original problem to that of the manifold. Additionally, under some restrictions, convergence results for the method used to solve the inequality constrained optimization problem can be carried over directly to our procedure.