Minimization of the pseudospectral abscissa of a matrix polynomial

Abstract

For a quadratic matrix polynomial dependent on parameters and a given tolerance epsilon > 0, the minimization of the epsilon-pseudospectral abscissa over the set of permissible parameter values is discussed, with applications in damping optimization and brake squeal reductions in mind. An approach is introduced that is based on nonsmooth and global optimization (or smooth optimization techniques such as BFGS if there are many parameters) equipped with a globally convergent crisscross algorithm to compute the epsilon-pseudospectral abscissa objective when the matrix polynomial is small. For the setting when the matrix polynomial is large, a subspace framework is introduced, and it is argued formally that it solves the minimization problem globally. The subspace framework restricts the parameter-dependent matrix polynomial to small subspaces and thus solves the minimization problem for such restricted small matrix polynomials. It then expands the subspaces using the minimizers for the restricted polynomials. The proposed approach makes the global minimization of the epsilon-pseudospectral abscissa possible for a quadratic matrix polynomial dependent on a few parameters and for sizes up to at least a few hundreds. This is illustrated in several examples originating from damping optimization.