On the attraction of Newton’s method to critical Lagrange multipliers

The attraction of dual trajectories of Newton’s method for the Lagrange system to critical Lagrange multipliers is analyzed. This stable effect, which has been confirmed by numerical practice, leads to the Newton–Lagrange method losing its superlinear convergence when applied to problems with irregular constraints. At the same time, available theoretical results are of “negative” character; i.e., they show that convergence to a noncritical multiplier is not possible or unlikely. In the case of a purely quadratic problem with a single constraint, a “positive” result is proved for the first time demonstrating that the critical multipliers are attractors for the dual trajectories. Additionally, the influence exerted by the attraction to critical multipliers on the convergence rate of direct and dual trajectories is characterized.