Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

Fabio Botelho

doi:10.20944/preprints202210.0091.v9

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preprints.org > computer science and mathematics > applied mathematics > doi: 10.20944/preprints202210.0091.v9

Preprint Article Version 9 Preserved in Portico This version is not peer-reviewed

Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

Fabio Botelho

Version 1 : Received: 24 September 2022 / Approved: 8 October 2022 / Online: 8 October 2022 (03:50:31 CEST)
Version 2 : Received: 11 October 2022 / Approved: 12 October 2022 / Online: 12 October 2022 (03:54:18 CEST)
Version 3 : Received: 24 October 2022 / Approved: 24 October 2022 / Online: 24 October 2022 (05:39:11 CEST)
Version 4 : Received: 24 November 2022 / Approved: 24 November 2022 / Online: 24 November 2022 (06:54:28 CET)
Version 5 : Received: 4 December 2022 / Approved: 5 December 2022 / Online: 5 December 2022 (02:57:53 CET)
Version 6 : Received: 7 December 2022 / Approved: 7 December 2022 / Online: 7 December 2022 (02:47:54 CET)
Version 7 : Received: 14 December 2022 / Approved: 14 December 2022 / Online: 14 December 2022 (06:25:21 CET)
Version 8 : Received: 16 December 2022 / Approved: 16 December 2022 / Online: 16 December 2022 (07:34:56 CET)
Version 9 : Received: 19 December 2022 / Approved: 20 December 2022 / Online: 20 December 2022 (02:32:43 CET)
Version 10 : Received: 4 January 2023 / Approved: 4 January 2023 / Online: 4 January 2023 (03:45:53 CET)
Version 11 : Received: 23 January 2023 / Approved: 25 January 2023 / Online: 25 January 2023 (04:21:30 CET)
Version 12 : Received: 29 January 2023 / Approved: 29 January 2023 / Online: 29 January 2023 (09:58:00 CET)

A peer-reviewed article of this Preprint also exists.

Botelho, F.S. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations. Mathematics 2022, 11, 63, doi:10.3390/math11010063. Botelho, F.S. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations. Mathematics 2022, 11, 63, doi:10.3390/math11010063.

Abstract

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. In the first sections, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section we present some numerical results concerning the generalized method of lines applied to a Ginzburg-Landau type equation.

Keywords

Duality principles; Generalized method of lines; Ginzburg-Landau type equations

Subject

Computer Science and Mathematics, Applied Mathematics

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Comments (1)

Comment 1

Received: 20 December 2022
Commenter: Fabio Botelho
Commenter's Conflict of Interests: Author

Comment: Dear Sir Editor

We have replaced one section by another we believe is better.

Also, we have included a section with new primal-dual formulations and related new convex formulations for the original primal problem.

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