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.