Modern Homotopy Methods in Optimization

Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new techniques have been successfully applied to solve Brouwer faced point problems, polynomial systems of equations, and discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements. This paper summarizes the theory of globally convergent homotopy algorithms for unconstrained and constrained optimization, and gives some examples of actual application of homotopy techniques to engineering optimization problems.