ИСТИНА

We discuss possible scenarios of behaviour of the dual part of the trajectory generated by a Newton-type methods (SQP, semismooth Newton methods for equation-based reformulations of Karush-Kuhn-Tucker systems) when applied to constrained optimization problems with nonunique multipliers associated to a solution (i.e., violating strict Mangasarian-Fromovitz constraint qualification). Among those scenarios are (a) failure of convergence of the dual sequence; (b) convergence to a so-called critical multiplier (which, in particular, violates the second-order sufficient condition for optimality), both satisfying and violating strict complementarity, which appears to be a typical scenario when critical multiplier exists; (c) convergence to a noncritical multiplier. The case of mathematical programs with complementarity constraints is also discussed. We illustrate each scenario with examples, and discuss consequences for the speed of convergence.