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 (). 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 (). 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 ().
Maddocks (, see also ) 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  in two ways. The first is obvious:  dealt with single constraints. In addition, in  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 .
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 (). We shall be using this fact throughout the paper.