A Survey of Probability-One Homotopy Methods for Engineering 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 globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, discretizations on nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and Galerkin approximations to nonlinear partial differential equations. This paper surveys the basic theory of globally convergent probability-one homotopy algorithms relevant to optimization, describes some computer algorithms and mathematical software, and applies homotopy theory to unconstrained optimization, constrained optimization, and global optimization of polynomial programs. In addition, two realistic engineering applications (optimal design of composite laminated plates and fuel-optimal orbital satellite maneuvers) are presented.