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We consider the model problem: given an affine matrix family $A(x)$ parameterized by a vector $x$, minimize the spectral abscissa (maximum real part of the eigenvalues) of $A(x)$, subject to bounds on $\|x\|$. Local minimizers of this class of problems typically have the following property: the spectral abscissa is not differentiable, in fact not even Lipschitz, at the minimizer, because of the presence of multiple eigenvalues. We give some explanation for this phenomenon; among our tools are optimality conditions derived using nonsmooth analysis. We also present a gradient bundle algorithm for approximating local minimizers, and we compare the results to those obtained using a Newton barrier method to minimize a related "robust" spectral abscissa. This work is joint with James Burke (U. Washington) and Adrian Lewis (U. Waterloo).