ИСТИНА

Critical multipliers are special Lagrange multipliers usually forming a thin subset in the set of all multipliers when the latter set is not a singleton. In particular, such multipliers necessarily violate the second-order sufficient optimality conditions. By now, there exists a convincing theoretical and numerical evidence of the following striking phenomena: dual sequences generated by Newton-type methods for optimality systems have a strong tendency to converge to critical multipliers when the latter exist, and moreover, this is precisely the reason for the lack of superlinear convergence rate, which is typical for problems with degenerate constraints. However, the existing theoretical results of this kind are far from giving a complete picture. First, all these results are ``negative'' by nature: they attempt to give a characterization of what would have happened in the case of convergence to a noncritical multiplier, showing that this scenario is in a sense unlikely. Clearly, this analysis must be complemented by results of a ``positive'' nature, demonstrating that the set of critical multipliers is indeed an attractor in some sense. Second, the exiting results rely on some questionable assumptions, and perhaps the most questionable one is asymptotic stabilization of the primal directions generated by Newtonian subproblems. Obtaining the first result on actual local convergence to a critical multiplier, and avoiding undesirable assumptions, are the main goals of this work.