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 del2L and also investigate 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 del2L is not positive definite on the null space of the active constraints.