Degenerate Nonlinear Programming with a Quadratic Growth Condition

We show that the quadratic growth condition and the Mangasarian-Fromovitz constraint qualiication imply that local minima of nonlinear programs are isolated stationary points. As a result, when started suuciently close to such points, an L1 exact penalty sequential quadratic programming algorithm will induce at least R-linear convergence of the iterates to such a local minimum. We construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the Mangasarian-Fromovitz constraint qualiication but for which no positive semideenite augmented Lagrangian exists. We present numerical results obtained using several nonlinear programming packages on this example, and discuss its implications for some algorithms.