First- and Second-Order Epi-Differentiability in Eigenvalue Optimization

The subject of this work is the first- and second-order sensitivity analysis of some spectral functions which are essential in eigenvalue optimization by the way of epi-differentiability. We show that the sum of the m largest eigenvalues of a real symmetric matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. We also prove that the mth largest eigenvalue function is twice epi-differentiable if and only if it ranks first in a group of equal eigenvalues. Finally, we derive chain rules and then we obtain optimality conditions for an important class of eigenvalue optimization problems