Summary: Second-order negative-curvature methods for box-constrained and
general constrained optimization
E. G. Birgin
J. M. Mart´inez
M. L. Schuverdt
May 13, 2008.
A Nonlinear Programming algorithm that converges to second-order stationary points is intro-
duced in this paper. The main tool is a second-order negative-curvature method for box-constrained
minimization of a certain class of functions that do not possess continuous second derivatives. This
method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar)
type. Convergence proofs under weak constraint qualifications are given. Numerical examples show-
ing that the new method converge to second-order stationary points in situations in which first-order
methods fail are exhibited.
Key words: Nonlinear programming, Augmented Lagrangians, global convergence, optimality con-
ditions, second-order conditions, constraint qualifications.
We are concerned with the general nonlinear programming problem with equality and inequality con-
straints. Most practical nonlinear optimization algorithms aim to encounter a local solution of the