Morse Programs: A Topological Approach to Smooth Constrained Optimization

The purpose of this paper is to give a geometrical answer to the question to the strong second order sufficiency conditions hold at any local minimum point for almost all nonlinear programs? Our idea is to reduce the nonlinear programming problem to a finite family of “well-behaved” nonlinear programs by perturbing the objective function in a linear fashion and perturbing the right-hand side of the constraints by adding a constant. Each of the “well-behaved” nonlinear programs will consist of minimizing a Morse function on a manifold with boundary, where the Morse function has no critical points on the boundary.