Morse Programs

Spingarn and Rockafellar [13] showed in a program

(Qvu) minimize {f(x) - uTx subject to g(x) < b + v}

where f : Rn approaches R, g: Rn approaches Rm; f in C2, g in Cn; u in Rn, v in Rm; n > m; that at any local minimum point x of (Qvu) the Jacobian matrix of g at x has full rank, strict complementary slackness holds and the second order sufficiency conditions hold, for almost every matrix{u v} in Rn x Rm (Lebesgue measure)

The purpose of this paper is to explicate the geometry underlying their work and to exploit this geometry in the generic analysis of constrained optimization problems. Namely we show that their work can be reduced to the study of minimizing a Morse function on a manifold with boundary.

We follow a classical tradition of first studying an equality constrained program and then reducing inequality constrained programs to a finite family of equality constrained programs, through the device of active (or binding) constraints.