ИСТИНА

For constrained optimization problems with nonunique Lagrange multiplier associated to a solution, we define the thin subclass of multipliers, called critical (and in particular, violating the second-order sufficient condition for optimality), and possessing some very special properties. Specifically, convergence to a critical multiplier appears to be a typical scenario of dual behaviour of primal-dual Newton-type methods when critical multipliers do exist. Moreover, along with the possible absence of dual convergence, attraction to critical multipliers is precisely the reason for slow primal convergence. On the other hand, critical multipliers turn out to have some special analytical stability properties: noncritical multipliers should not be expected to be stable subject to parametric perturbations of optimality systems.