Todd MunsonSenior Fellow
My primary research focus is algorithms and applications of optimization and complementarity. Most recently I have been working on utilizing constrained nonlinear optimization techniques to compute mountain passes, critical points where the Hessian has exactly one negative eigenvalue. Mountain passes are of interest, for example, in computational chemistry where they correspond to transition states for chemical reactions. I have also been working with on an application of optimization to the r-refinement problem, a large nonlinear, nonconvex optimization problem that can cause grief for many general-purpose methods.
I have also worked on several generalized Newton methods for solving complementarity problems. PATH is based on the normal map formulation of complementarity problems. At each iteration, a linear complementarity problem is solved to calculate a direction. The code is globalized by using the Fischer-Burmeister merit function. The method is enhanced with preprocessing techniques that remove variables and constraints from the given problem, and uncover polyhedral variational inequality structure; nonmontone search criteria; and crashing methods to identify good active sets. Parallel versions of the semismooth methods that I wrote are available in TAO, the Toolkit for Advanced Optimization.
Finally, I have worked on special-purpose algorithms for solving support vector machine and mesh shape-quality optimization problems. The structure of the linear support vector machine problem enabled us to write tailored linear algebra in interior-point and semismooth algorithms so that we can solve extremely large instances (on the order of 60-100 million degrees of freedom).