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Journal of Convex Analysis 23 (2016), No. 2, 425--459
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

Jean Charles Gilbert
INRIA Paris, 2 Rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France

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, quasi-global error bound, shifted constraint.

MSC: 49M27, 49M29, 65K05, 90C05, 90C06, 90C20, 90C25.

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