We consider an augmented Lagrangian algorithm for minimizing a convex quadratic function subject to linear inequality constraints. Linear optimization is an important particular instance of this problem. We show that, provided the augmentation parameter is large enough, the constraint value converges globally linearly to zero. This property is proven by establishing first a global radial Lipschitz property of the reciprocal of the dual function subgradient. It is also a consequence of the proximal interpretation of the method. No strict complementarity assumption is needed. The result is illustrated by numerical experiments and algorithmic implications are discussed.