A primal-dual augmented Lagrangian and log-barrier penalty algorithm for nonlinear optimization
Résumé
We propose a new primal-dual algorithm for solving nonlinearly constrai- ned minimization problems. This is a Newton-like method applied to a per- turbation of the optimality system that follows from a reformulation of the initial problem by introducing an augmented Lagrangian and a log-barrier penalty to handle both equality and bound constraints. Two kinds of itera- tions are used. The outer iterations at which the different parameters, such as the Lagrange multipliers and the penalty parameters, are updated. The inner iterations to get a sufficient decrease of a given primal-dual penalty function. Both iterations use the same kind of coefficient matrix and the corresponding linear system is solved by means of a symmetric indefinite factorization including an inertia-controlling technique. The globalization is performed by means of a line search strategy on a primal-dual merit func- tion. An important aspect of this approach is that, by a choice of suitable update rules of the parameters, the algorithm reduces to a regularized New- ton method applied to a sequence of optimality systems derived from the original problem. The global convergence and the asymptotic properties of the algorithm are presented. In particular, we show that the algorithm is q-superlinear convergent. In addition, this method is able to solve the well known example of Wachter and Biegler, for which some interior point methods based on a line search strategy fail.