  We prove an extension of Yuan's Lemma to more than two matrices, as long as
the set of matrices has rank at most 2. This is used to generalize the main
result of [A. Baccari and A. Trad. On the classical necessary second-order
optimality conditions in the presence of equality and inequality constraints.
SIAM J. Opt., 15(2):394--408, 2005], where the classical necessary second-order
optimality condition is proved under the assumption that the set of Lagrange
multipliers is a bounded line segment. We prove the result under the more
general assumption that the hessian of the Lagrangian evaluated at the vertices
of the Lagrange multiplier set is a matrix set with at most rank 2. We apply
the results to prove the classical second-order optimality condition to
problems with quadratic constraints and without constant rank of the jacobian
matrix.
