Abstract:

We consider spectral functions f o lambda, where f is any permutation-invariant mapping from C^n to R, and lambda is the eigenvalue map from C^{n X n} to C^n, ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f o lambda is the spectral abscissa, while if f is "maximum modulus", then f o lambda is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998), which is particularly well suited to the variational analysis of non-Lipschitz spectral functions. We derive a number of necessary conditions for subgradients of spectral functions. For the spectral abscissa, we give both necessary and sufficient conditions for subgradients, and precisely identify the case where subdifferential regularity holds. We conclude by introducing the notion of semistable programming: minimizing a linear function of a matrix subject to linear constraints, together with the constraint that the eigenvalues of the matrix all lie in the right half-plane or on the imaginary axis. This is a generalization of semidefinite programming for non-Hermitian matrices. Using our analysis, we derive a necessary condition for a local minimizer of a semistable program, and give a generalization of the complementarity condition familiar from semidefinite programming.