Constrained Optimisation and Morse Theory

In classical Morse theory the number and type (index) of critical points of a smooth function on a manifold are related to topological invariants of that manifold through the Morse inequalities. There the index of a critical point is the number of negative eigenvalues that the Hessian matrix has on that tangent plane. Here deenitions of \critical point" and \index" are given that are suitable for functions on fx 2 M j gi(x) 0; i = 1; : : : ; mg. These deenitions are based on Kuhn{Tucker theory of constrained optimization. The Morse inequalities are proven for this new situation. These results may be extended to smooth manifolds with corners.