Globally Convergent Homotopy Methods for Large Scale Engineering Optimization

Abstract

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 globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, discretizations of non- linear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and finite difference, collocation, and Galerkin approximations to nonlinear partial differential equations. This paper surveys the basic theory of globally convergent homotopy algorithms, describes some computer algorithms and mathematical software, applies homotopy theory to unconstrained and constrained optimization, and presents two realistic engineering applications (optimal design of composite laminated plates and fuel-optimal orbital satellite maneuvers ).