Let \(\overline{\mathbb R}:= \mathbb R \cup \{\infty\}\) denote the set of the extended real numbers. Given a function \(f:\mathbb C \rightarrow \overline{\mathbb R}\), the spectral max function \(\mathsf{f}:\mathbb C^{n\times n} \rightarrow \overline{\mathbb R}\) generated by \(f\) is \[ \mathsf{f}(X):=\max\{f(\lambda)\mid \lambda \in \mathbb C \;\text{and} \;\det(\lambda I - X) = 0\}. \]
Two important spectral max functions are the spectral abscissa and spectral radius which are generated by \(f(\cdot) = \operatorname{Re}(\cdot)\) and \(f(\cdot) = |\cdot|\) respectively. Let \(\mathbb{P}^n\) denote the set of complex polynomials with degree equal or less than \(n\). The polynomial root max function generated by \(f\) is the mapping \(\mathbf{f}:\mathbb{P}^n \rightarrow \overline{\mathbb R}\) defined by \[ \mathbf{f}(p):=\max\{f(\lambda)\mid \lambda \in \mathbb C \;\text{and} \;p(\lambda) = 0\}. \] We say \(\mathbf{f}\) is convexly generated if \(f\) is proper, convex, and lsc.
In [Found. Comput. Math. 1, No. 2, 205–225 (2001; Zbl 0994.15022)], J. V. Burke et al. characterized the regular subdifferential of the spectral abscissa and showed that the spectral abscissa is subdifferentially regular if all active eigenvalues are nonderogatory. In this paper, the authors extend the Burke-Overton variational analysis results for the spectral abscissa mapping to the class of convexly spectral max functions. It was also shown that subdifferential regularity occurs if and only if all active eigenvalues are nonderogatory.
Two different approaches are applied. The first approach uses the Arnold form [V. I. Arnol’d, Russ. Math. Surv. 26, No. 2, 29–43 (1972; Zbl 0259.15011); translation from Usp. Mat. Nauk 26, No. 2(158), 101–114 (1971)] to derive a representation for the subdifferential and establish the subdifferential regularity of convexly generated spectral max functions at matrices with nonderogatory active eigenvalues. The second approach uses the underlying matrix structures to characterize the regular subgradients of convexly generated spectral max functions without assuming nonderogatory active eigenvalues.