The U-Lagrangian of the maximum eigenvalue function

In this paper we apply the U-Lagrangian theory to the maximum eigenvalue fu nction lambda(1) and to its precomposition with affine matrix-valued mappin gs. We first give geometrical interpretations of the U-objects that we intr oduce. We also show that the U-Lagrangian of lambda(1) has a Hessian which can be explicitly computed; the second-order development of the U-Lagrangia n provides a second-order development of lambda(1) along a characteristic s mooth manifold: the set of symmetric matrices whose maximal eigenvalues hav e a fixed multiplicity. The same results can be obtained when we precompose lambda(1) with an affine matrix-valued mapping A, provided that this mappi ng satisfies a regularity condition (transversality condition). We show tha t the Hessian of the U-Lagrangian of lambda(1) circle A coincides with the reduced Hessian encountered in sequential quadratic programming. Finally, w e use the U-Lagrangian to derive second-order algorithms for minimizing lam bda(1) circle A.