ИСТИНА

According to some recently obtained results, multiplier methods appear to have very nice local convergence properties even for optimization problems with nonunique Lagrange multipliers associated to a solution. Specifically, employing the special (but very natural) residual-based rule to control the penalty parameter in the augmented Lagrangian, local uperlinear convergence can be established without any constraint qualifications provided the primal-dual starting point is close enough to a primal-dual solution satisfying the standard econd-order sufficient optimality condition. These results put into the foreground the question about possible and typical scenarios of dual behavior of multiplier methods. One relevant statement of the question is the following: assuming convergence of the primal sequence, what are the possible accumulation points of the dual trajectory? In this work, we attempt to answer this question: we demonstrate that dual accumulation points possess some very special properties. This analysis is strongly related to the theory of critical Lagrange multipliers which turn out to play a crucial role for the dual behavior of the Newton-type methods for optimization problems with degenerate constraints. We demonstrate that accumulation points of dual sequences of multiplier methods must satisfy the second-order necessary optimality conditions, or be critical, or both. We also study how different scenarios of dual behavior affect the convergence rate. Special attention is paid to the cases of finite termination, which are quite typical when full degeneracy occurs, and which also lead to some special properties of dual termination points. Theoretical analysis presented in this work is supported by examples and numerical results for randomly generated test problems and for test problems from DEGEN test collection.