Suppose that *I* and , *i*=1, , *n* are functions with at
least two Fréchet derivatives defined on a Hilbert space . Suppose
that satisfies the *n* constraints . The first
order necessary condition for *I* to have a strong local minimum
at subject to these constraints is that there exist Lagrange
multipliers so that is identically
zero ([1]). Here *I*' and are Fréchet derivatives
evaluated at *y*. We will assume that this condition is satisfied. A
second order necessary condition is that be
non-negative definite on the tangent space to the constraint manifold at
*y* ([3]). In order to prove that *y* is a strong local
constrained minimum (i.e., that *I*(*y*+*h*)>*I*(*y*) for all non-zero *h* of
sufficiently small magnitude satisfying , *i*=1, ,
*n*), a first step is often to show that is
positive definite on the tangent space. Even in the unconstrained case,
however, this is not by itself sufficient ([1]).

Maddocks ([3], see also [4]) derived
sufficient conditions for to be positive
definite on the space tangent to the constraint manifold. These conditions
involve the spectrum of the bounded, self-adjoint operator *A* defined by
, where is
the inner product of . In this paper, I will start with Maddocks'
criteria to obtain sufficient conditions for constrained strong local
extrema. This paper generalizes [6] in two ways. The first is
obvious: [6] dealt with single constraints. In addition,
in [6] the assumption was made that *A* was invertible. We will
weaken this assumption in two different ways. In Theorem 2.1,
*A* might fail to be invertible if one of the non-positive eigenvalues
happens to equal zero. On the other hand, in Theorem 3.1 the
kernel of *A* is treated separately. The result of this is that when *A*
is not invertible, the two theorems yield slightly different criteria. For
a (one constraint) example in which *A* is singular, see [5].

Throughout this paper, the kernel of *A* will be finite dimensional. This
is certainly necessary for *A* to be positive definite on the tangent
space, which is of finite co-dimension. However, I won't state this as a
separate assumption, since it will be implied by the hypotheses of the two
theorems. In both of the theorems of this paper, 0 is an isolated point of
the spectrum of *A*. Because of this, the range of *A* (which I shall call
) is closed, so that ([2]).
We shall be using this fact throughout the paper.

Tue Aug 20 10:23:22 CDT 1996