A Monomial Method for Nonlinear Optimization in Astrodynamics

This paper introduces a monomial method for rapidly solving a variety of nonlinear optimization problems astrodynamics. Using differential algebra to compute Taylor expansions of given nonlinear functions, the result is a function of multivariate monomials up to the order of truncation. These monomials are interrelated. We use this fact to approximate nonlinear functions in Euclidean space by linear functions on a Riemannian manifold, wherein the problem can be solved with speed and stability, leveraging numerical schemes for optimization on manifolds. The method could prove advantageous in contexts requiring the repeated solution of similar optimization problems, for which the expansion can be reused, or for particularly tricky optimization problems where a combination of rapid sampling and gradient-based methods need to be used. In this introductory work we lay the foundations and explore two simple example applications: 1) computing periodic orbits, and 2) computing/continuing optimal impulsive transfers.