On Minimizing the Special Radius of a Nonsymmetric Matrix Function: Optimality Conditions and Duality Theory

Abstract

Let $A( x )$ be a nonsymmetric real matrix affine function of a real parameter vector $x \in \mathcal{R}^m $, and let $\rho ( x )$ be the spectral radius of $A( x )$. The article addresses the following question: Given $x_0 \in \mathcal{R}^m $, is $\rho ( x )$ minimized locally at $x_0 $, and, if not, is it possible to find a descent direction for $\rho ( x )$ from $x_0 $? If any of the eigenvalues of $A( x_0 )$ that achieve the maximum modulus $\rho ( x_0 )$ are multiple, this question is not trivial to answer, since the eigenvalues are not differentiable at points where they coalesce. In the symmetric case, $A( x ) = A( x )^T $ for all $x,\rho ( x )$ is convex, and the question was resolved recently by Overton following work by Fletcher and using Rockafellar’s theory of subgradients. In the nonsymmetric case $\rho ( x )$ is neither convex nor Lipschitz, and neither the theory of subgradients nor Clarke’s theory of generalized gradients is applicable. A new necessary and sufficient condition is given for...