On solving the condensed KKT system with application to design optimization

We consider a general nonlinear optimization problem subject to equality and inequality constraints. Suppose that the first and second derivatives of the objective and constraint functions are available. The optimization problem is solved by the Lagrange multiplier approach and the interior-point method. The inequality constraints are introduced as logarithmic terms in a family of optimization problems depending on a positive barrier parameter that asymptotically goes to zero. Applying the Karush-Kuhn-Tucker (KKT) first-order necessary optimality conditions we are faced with a nonlinear equation for which the Newton method is used. The resulting primal-dual system of linear equations can be symmetrized and transformed into a problem of smaller dimension. The condensed KKT matrix is symmetric but indefinite and increasingly ill-conditioned as the optimization proceeds. To find an optimal solution satisfying the perturbed optimality conditions we apply an iterative solver based on the null space approach in conjunction with suitable preconditioners. For the stepsize selection we use an augmented Lagrangian merit function. The method is illustrated on a PDE-constrained shape optimization problem with box constraints for the design parameters.