ИСТИНА

It is known that when the set of Lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton, this set may contain so-called critical multipliers. This special subset of Lagrange multipliers defines, to a great extent, the stability pattern of solution in question subject to perturbations of the problem data, and the behavior of Newton-type methods. Criticality of a Lagrange multiplier can be equivalently characterized by the absence of the local Lipschitzian error bound in terms of the natural residual of the optimality system, and this view of criticality serves as a basis for extension of this concept to general nonlinear equations (not necessarily with primal-dual optimality structure). This extension is of special interest when the solution in question can be nonisolated (and in particular, singular). We demonstrate that under some natural condition, various Newton-type methods have large domains of attraction to critical solutions. Apart from other things, the new results obtained on this way give a new understanding of critical Lagrange multipliers and their properties, from some more general principles.