Minimization of the Pseudospectral Abscissa of a Quadratic Matrix Polynomial

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 criss-cross algorithm to compute the $\epsilon$-pseudospectral abscissa objective when the matrix polynomial is of small size. 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 on several examples originating from damping optimization.