Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption

Izmailov, A.F.; Kurennoy, A.S.; Solodov, M.V.

Статья опубликована в высокорейтинговом журнале

Информация о цитировании статьи получена из Web of Science, Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 2 апреля 2015 г.

Авторы: Izmailov A.F., Kurennoy A.S., Solodov M.V.
Журнал: Computational Optimization and Applications
Том: 60
Номер: 1
Год издания: 2015
Издательство: Springer Nature
Местоположение издательства: Switzerland
Первая страница: 111
Последняя страница: 140
DOI: 10.1007/s10589-014-9658-8
Аннотация: We present a local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local Q-superlinear convergence is established under the appropriate control of the penalty parameter values. For optimization problems, we demonstrate in addition local Q-linear convergence for sufficiently large fixed penalty parameters. Both exact and inexact versions of the method are considered. Contributions with respect to previous state-of-the-art analyses for equality-constrained problems consist in the extension to the variational setting, in using the weaker noncriticality assumption instead of the usual second-order sufficient optimality condition (SOSC), and in relaxing the smoothness requirements on the problem data. In the context of optimization problems, this gives the first local convergence results for the augmented Lagrangian method under the assumptions that do not include any constraint qualifications and are weaker than the SOSC. We also show that the analysis under the noncriticality assumption cannot be extended to the case with inequality constraints, unless the strict complementarity condition is added (this, however, still gives a new result).
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Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumptionстатья

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