Large-Scale Minimization of the Pseudospectral Abscissa

This work concerns the minimization of the pseudospectral abscissa of a matrix-valued function dependent on parameters analytically. The problem is motivated by robust stability and transient behavior considerations for a linear control system that has optimization parameters. We describe a subspace procedure to cope with the setting when the matrix-valued function is of large size. The proposed subspace procedure solves a sequence of reduced problems obtained by restricting the matrix-valued function to small subspaces, whose dimensions increase gradually. It possesses desirable features such as the global convergence of the minimal values of the reduced problem to the minimal value of the original problem, and a superlinear convergence exhibited by the decay in the errors of the minimizers of the reduced problems. In mathematical terms, the problem we consider is a large-scale nonconvex minimax eigenvalue optimization problem such that the eigenvalue function appears in the constraint of the inner maximization problem. Devising and analyzing a subspace framework for the minimax eigenvalue optimization problem at hand with the eigenvalue function in the constraint require special treatment that makes use of a Lagrangian and dual variables. There are notable advantages in minimizing the pseudospectral abscissa over maximizing the distance to instability or minimizing the $\mathcal{H}_\infty$ norm; the optimized pseudospectral abscissa provide quantitative information about the worst-case transient behavior, and the initial guesses for the parameter values to optimize the pseudospectral abscissa can be arbitrary, unlike the case to optimize the distance to instability and $\mathcal{H}_\infty$ norm that would normally require initial guesses yielding asymptotically stable systems.