To solve complicated nonlinear optimization problems arising in geometric and real-world scenarios, usually iterative methods are used. However, these methods often require a good initialization to guarantee fast convergence, which is not always available. Instead, Riemannian optimization uses local methods consisting of linear steps to solve optimization problems whose constraint set is a smooth manifold. Since a linear step along some descent direction generally leaves the set of constraints, retraction maps are used to approximate the exponential map and return to the manifold. For implicitly-defined manifolds, suitable retraction maps are difficult to find. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean closest point retraction for any implicitly-defined Euclidean submanifold, and prove convergence results. In addition, we show how this approach works in more general settings such as hyperbolic manifolds, statistical models with the maximum likelihood metric and even on algebraic varieties, which may contain singularities.

(This is joint work with Alexander Heaton)