A Note on the Augmented Hessian Whenthe Reduced Hessian is Semide

Certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semideenite programming. Let H be an n n symmetric matrix, A an m n matrix, and Z a basis for the null space of A. (In the context of optimization, H is the Hessian of a smooth function and A is the Jacobian of a set of constraints.) When the reduced Hessian Z T HZ is positive deenite, augmented Lagrangian methods rely on the known existence of a nite such that, for all > , the augmented Hessian H +A T A is positive deenite. In this note we analyze the case when Z T HZ is positive semideenite, i.e. singularity is allowed, and show that the situation is more complicated. In particular, we give a simple necessary and suucient condition for the existence of a nite so that H + A T A is positive semideenite for. A corollary of our result is that if H is nonsingular and indeenite while Z T HZ is positive semideenite and singular, no such exists.