The -Lagrangian of the Maximum Eigenvalue Function

In this paper we apply the -Lagrangian theory to the maximum eigenvalue function and to its precomposition with affine matrix-valued mappings. We first give geometrical interpretations of the -objects that we introduce. We also show that the -Lagrangian of has a Hessian which can be explicitly computed; the second-order development of the -Lagrangian provides a second-order development of along a characteristic smooth manifold: the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity. The same results can be obtained when we precompose with an affine matrix-valued mapping A, provided that this mapping satisfies a regularity condition ({ transversality condition}). We show that the Hessian of the -Lagrangian of ∘ A coincides with the reduced Hessian encountered in sequential quadratic programming. Finally, we use the -Lagrangian to derive second-order algorithms for minimizing ∘ A.