Using the KKT matrix in an augmented Lagrangian SQP method for sparse constrained optimization

The augmented Lagrangian SQP subroutine OPALQP was originally designed for small-to-medium sized constrained optimization problems in which the main calculation on each iteration, the solution of a quadratic program, involves dense, rather than sparse, matrices. In this paper, we consider some reformulations of OPALQP which are better able to take advantage of sparsity in the objective function and constraints.The modified versions of OPALQP differ from the original in using sparse data structures for the Jacobian matrix of constraints and in replacing the dense quasi-Newton estimate of the inverse Hessian of the Lagrangian by a sparse approximation to the Hessian. We consider a very simple sparse update for estimating ² L and also investigate the benefits of using exact second derivatives, noting in the latter case that safeguards are needed to ensure that a suitable search direction is obtained when ² L is not positive definite on the null space of the active constraints.The authors are grateful to John Reid and Nick Gould of the Rutherford Appleton Laboratory for a number of helpful and interesting discussions. Thanks are also due to Laurence Dixon for comments which led to the clarification of some parts of the paper.This work has been partly supported by a CAPES Research Studentship funded by the Brazilian Government.