
Journal of Convex Analysis 23 (2016), No. 2, 425459 Copyright Heldermann Verlag 2016 How the Augmented Lagrangian Algorithm Can Deal with an Infeasible Convex Quadratic Optimization Problem Alice Chiche Artelys France, 12 rue du Quatre Septembre, 75002 Paris, France Alice.Chiche@artelys.com Jean Charles Gilbert INRIA Paris, 2 Rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France JeanCharles.Gilbert@inria.fr This paper analyses the behavior of the augmented Lagrangian algorithm when it deals with an infeasible convex quadratic optimization problem. It is shown that the algorithm finds a point that, on the one hand, satisfies the constraints shifted by the smallest possible shift that makes them feasible and, on the other hand, minimizes the objective on the corresponding shifted constrained set. The speed of convergence to such a point is globally linear, with a rate that is inversely proportional to the augmentation parameter. This suggests us a rule for determining the augmentation parameter that aims at controlling the speed of convergence of the shifted constraint norm to zero; this rule has the advantage of generating bounded augmentation parameters even when the problem is infeasible. Keywords: Augmented Lagrangian algorithm, augmentation parameter update, closest feasible problem, convex quadratic optimization, feasible shift, global linear convergence, infeasible problem, proximal point algorithm, quasiglobal error bound, shifted constraint. MSC: 49M27, 49M29, 65K05, 90C05, 90C06, 90C20, 90C25. [ Fulltextpdf (281 KB)] for subscribers only. 