An Inequality Constrained SL/QP Method for Minimizing the Spectral Abscissa

TL;DR

This paper introduces a new inequality constrained sequential linear and quadratic programming method for minimizing the spectral abscissa, addressing a challenging eigenvalue optimization problem crucial for control system stabilization.

Contribution

The paper proposes a novel inequality constrained SL/QP algorithm specifically designed for spectral abscissa minimization, handling nonconvex, nonsmooth, and non-Lipschitz objectives.

Findings

Algorithm effectively finds local minima in spectral abscissa problems.

Numerical results show improved performance over existing methods.

Method enhances stability analysis in control systems.

Abstract

We consider a problem in eigenvalue optimization, in particular finding a local minimizer of the spectral abscissa - the value of a parameter that results in the smallest value of the largest real part of the spectrum of a matrix system. This is an important problem for the stabilization of control systems. Many systems require the spectra to lie in the left half plane in order for them to be stable. The optimization problem, however, is difficult to solve because the underlying objective function is nonconvex, nonsmooth, and non-Lipschitz. In addition, local minima tend to correspond to points of non-differentiability and locally non-Lipschitz behavior. We present a sequential linear and quadratic programming algorithm that solves a series of linear or quadratic subproblems formed by linearizing the surfaces corresponding to the largest eigenvalues. We present numerical results comparing the algorithms to the state of the art.