On the analytical and numerical stability of critical Lagrange multipliers

If the constraint qualification does not hold at a stationary point of a constrained optimization problem, then the corresponding Lagrange multiplier may not be unique. Moreover, in the set of multipliers, one can select special (so-called critical) multipliers possessing certain specific properties that are lacking in the other multipliers. In particular, it is the critical multipliers that are usually stable with respect to small perturbations, and it is the critical multipliers that attract trajectories of Newton's method as applied to the Lagrange system of equations. The present paper is devoted to an analysis of these issues.