We are able to perform direct search methods over general Lipschitz manifolds. The manifold will typically arise from Lipschitz equality constraints. The method can also be practically employed on abstract Riemannian manifolds.

This is the first general method for applying direct search methods to equality constrained problems. Direct search methods have been employed for over 40 years. Given their recent advances, extensions to equality constrained problems is an important breakthrough.

A direct search method over the Grassmann manifold is needed to solve the Whitney embedding problem. The Whitney embedding algorithm is itself current state-of-the-art for dimensionality reduction of manifold-valued data. The ability to solve this embedding problem exactly is an important advancement over previous techniques.

Another application of optimization on manifolds is Inverse protein design. The protein folding/design problem is a large scale optimization problem. It can be stated as a multidisciplinary design optimization problem where finding derivative information may be prohibitively expensive or impossible. Being able to handle equality constraints is crucial in order to deal with this problem in order to relate the design variables to the system variables.

• A Pseudo-isometric Embedding Algorithm
• Direct Search Methods over Lipshitz Manifolds
• Approximate Inverse and Implicit Functions Using an M/Nset Variation
• Equality Constraints, Riemannian Manifolds and Direct Search Methods
• Direct Search Algorithms over Riemannian Manifolds