The paper proposes a primal-dual algorithm for solving an equality constrained minimization problem. The algorithm is a Newton-like method applied to a sequence of perturbed optimality systems that follow naturally from the quadratic penalty approach. This work is first motivated by the fact that a primal-dual formulation of the quadratic penalty provides a better framework than the standard primal form. This is highlighted by strong convergence properties proved under standard assumptions. In particular, it is shown that the usual requirement of solving the penalty problem with a precision of the same size as the perturbation parameter, can be replaced by a much less stringent criterion, while guaranteeing the superlinear convergence property. A second motivation is that the method provides an appropriate regularization for degenerate problems with a rank deficient Jacobian of constraints. The numerical experiments clearly bear this out. Another important feature of our algorithm is that the penalty parameter is allowed to vary during the inner iterations, while it is usually kept constant. This alleviates the numerical problem due to ill-conditioning of the quadratic penalty, leading to an improvement of the numerical performances.
Researh Report, University of Limoges (France), Research Institute XLIM, May 2012