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Perturbation Method and Regularization of the Lagrange Principle in Nonlinear Constrained Optimization Problems.

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Title: Perturbation Method and Regularization of the Lagrange Principle in Nonlinear Constrained Optimization Problems.
Authors: Sumin, M. I.1 (AUTHOR) m.sumin@mail.ru
Source: Computational Mathematics & Mathematical Physics. Dec2024, Vol. 64 Issue 12, p2823-2844. 22p.
Subjects: Mathematical forms, Constrained optimization, Computational mathematics, Lagrange multiplier, Subgradient methods
Abstract: The Lagrange principle in nondifferential form is regularized for a nonlinear (nonconvex) constrained optimization problem with an operator equality constraint in a Hilbert space. The feasible set of the problem belongs to a complete metric space, and the existence of its solution is not assumed a priori. The equality constraint involves an additive parameter, so a "nonlinear version" of the perturbation method can be used to study the problem. The regularized Lagrange principle is used mainly for the stable generation of generalized minimizing sequences (GMSes) in the considered nonlinear problem. It can be treated as a GMS-generating (regularizing) operator that takes each set of initial data of the problem to a subminimal (minimal) of its regular augmented Lagrangian corresponding to this set with the dual variable generated by the Tikhonov stabilization procedure for the dual problem. The augmented Lagrangian is completely determined by the form of "nonlinear" subdifferentials of the lower semicontinuous and generally nonconvex value function, which is regarded as a function of parameters of the problem. As such subdifferentials, we use the proximal subgradient and the Fréchet subdifferential, which are well known in nonsmooth (nonlinear) analysis. The regularized Lagrange principle overcomes the ill-posedness of its classical counterpart and can be treated as a regularizing algorithm, thus providing the theoretical basis for the development of stable methods for practically solving nonlinear constrained optimization problems. [ABSTRACT FROM AUTHOR]
Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: The Lagrange principle in nondifferential form is regularized for a nonlinear (nonconvex) constrained optimization problem with an operator equality constraint in a Hilbert space. The feasible set of the problem belongs to a complete metric space, and the existence of its solution is not assumed a priori. The equality constraint involves an additive parameter, so a "nonlinear version" of the perturbation method can be used to study the problem. The regularized Lagrange principle is used mainly for the stable generation of generalized minimizing sequences (GMSes) in the considered nonlinear problem. It can be treated as a GMS-generating (regularizing) operator that takes each set of initial data of the problem to a subminimal (minimal) of its regular augmented Lagrangian corresponding to this set with the dual variable generated by the Tikhonov stabilization procedure for the dual problem. The augmented Lagrangian is completely determined by the form of "nonlinear" subdifferentials of the lower semicontinuous and generally nonconvex value function, which is regarded as a function of parameters of the problem. As such subdifferentials, we use the proximal subgradient and the Fréchet subdifferential, which are well known in nonsmooth (nonlinear) analysis. The regularized Lagrange principle overcomes the ill-posedness of its classical counterpart and can be treated as a regularizing algorithm, thus providing the theoretical basis for the development of stable methods for practically solving nonlinear constrained optimization problems. [ABSTRACT FROM AUTHOR]
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  Group: Ab
  Data: <i>Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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