  In this paper we deal with optimality conditions that can be verified by a
nonlinear optimization algorithm, where only a single Lagrange multiplier is
avaliable. In particular, we deal with a conjecture formulated in [R. Andreani,
J.M. Martinez, M.L. Schuverdt, "On second-order optimality conditions for
nonlinear programming", Optimization, 56:529--542, 2007], which states that
whenever a local minimizer of a nonlinear optimization problem fulfills the
Mangasarian-Fromovitz Constraint Qualification and the rank of the set of
gradients of active constraints increases at most by one in a neighborhood of
the minimizer, a second-order optimality condition that depends on one single
Lagrange multiplier is satisfied. This conjecture generalizes previous results
under a constant rank assumption or under a rank deficiency of at most one. In
this paper we prove the conjecture under the additional assumption that the
Jacobian matrix has a smooth singular value decomposition, which is weaker than
previously considered assumptions. We also review previous literature related
to the conjecture.
