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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. ᮊ 1999 Academic Press

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