Ernesto G. Birgin (egbirgin@ime.usp.br)
Recent advances in numerical methods for nonlinear programming

This special session focuses on recent advances in the field of
numerical methods and algorithms for nonlinear programing. It contains
six contributions made by eleven different researchers. It highlights
recently introduced methods for derivative-free optimization,
linearly-constrained minimization, two Augmented-Lagrangian-based
methods (one with convergence to second-order stationary points and
other for deterministic global optimization of NLP problems), a method
based on trust regions and filter for NLP problems with equalities and
bounds, and the application of a projected-gradient-like algorithm to
NLP formulations of the eigenvalue complementarity problem.

In the first talk, a sample path method for derivative free
optimization, based on the construction of a quadratic model of the
objective function being minimized, is presented. The second talk
deals with an active-set strategy for linearly constrained
optimization. The method, named Partial Spectral projected Gradient,
modifies and generalizes recent box-constraint optimization
algorithms. Then, an Augmented Lagrangian method that converges to
second-order stationary points is presented. Its 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. Also based on the Augmented Lagrangian
framework, a deterministic global optimization method for NLP problems
is considered in the next presentation. Following that, based on trust
regions and filters, a nonmonotone method for nonlinear programming
problems with equality constraints and bounds is presented. Finally,
NLP formulations of the symmetric eigenvalue complementarity problem
are considered. The unsymmetric case is also tackled. Projected
gradient and enumerative algorithms are considered for their
resolution, respectively.