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Introduction.

Suppose that I and tex2html_wrap_inline434 , i=1, tex2html_wrap_inline438 , n are functions with at least two Fréchet derivatives defined on a Hilbert space tex2html_wrap_inline442 . Suppose that tex2html_wrap_inline464 satisfies the n constraints tex2html_wrap_inline468 . The first order necessary condition for I to have a strong local minimum at tex2html_wrap_inline464 subject to these constraints is that there exist Lagrange multipliers tex2html_wrap_inline474 so that tex2html_wrap_inline476 is identically zero ([1]). Here I' and tex2html_wrap_inline480 are Fréchet derivatives evaluated at y. We will assume that this condition is satisfied. A second order necessary condition is that tex2html_wrap_inline484 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 tex2html_wrap_inline494 , i=1, tex2html_wrap_inline498 , n), a first step is often to show that tex2html_wrap_inline484 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 tex2html_wrap_inline484 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 tex2html_wrap_inline508 , where tex2html_wrap_inline510 is the inner product of tex2html_wrap_inline442 . 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 tex2html_wrap_inline532 ) is closed, so that tex2html_wrap_inline534 ([2]). We shall be using this fact throughout the paper.


next up previous
Next: Non-positive eigenvalues Up: Sufficient Conditions for Multiply Previous: Sufficient Conditions for Multiply

Tom Vogel
Tue Aug 20 10:23:22 CDT 1996