The Space Decomposition Theory for a Class of Semi-Infinite Maximum Eigenvalue Optimizations

Abstract

We study optimization problems involving eigenvalues of symmetric matrices. We present a nonsmooth optimization technique for a class of nonsmooth functions which are semi-infinite maxima of eigenvalue functions. Our strategy uses generalized gradients and $𝒰𝒱$ space decomposition techniques suited for the norm and other nonsmooth performance criteria. For the class of max-functions, which possesses the so-called primal-dual gradient structure, we compute smooth trajectories along which certain second-order expansions can be obtained. We also give the first- and second-order derivatives of primal-dual function in the space of decision variables $R^m$ under some assumptions.