Nonsmooth optimization problems involve objectives or constraints which are not differentiable everywhere. The term structured nonsmooth optimization means that the specific nonsmooth nature of the problem is known. One especially interesting class repeatedly arises in many applications: problems involving eigenvalues of matrices. The proposed work primarily focuses on various aspects of eigenvalue optimization, including semidefinite programming, quasiconvex eigenvalue optimization, and non-Lipschitz eigenvalue optimization. The goal is fourfold: the development of fast, robust numerical algorithms; analysis of theoretical questions concerning optimality conditions and algorithm convergence; development of software which can be used by the general scientific community; application to the solution of important interesting problems which arise in practice. Application areas of particular interest include combinatorial optimization, robust control and structural analysis. ***
