{"title":"适用于局部和全局非凸变分优化的对偶原理及相关凸对偶公式","authors":"Fabio Silva Botelho","doi":"10.1515/nleng-2022-0343","DOIUrl":null,"url":null,"abstract":"Abstract This article develops duality principles, a related convex dual formulation and primal dual formulations suitable for the local and global optimization of non convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg–Landau type equation. Other applications include primal dual variational formulations for a Burger’s type equation and a Navier–Stokes system. We emphasize the novelty here is that the first dual variational formulation developed is convex for a primal formulation which is originally non-convex. Finally, we also highlight the primal dual variational formulations presented have a large region of convexity around any of their critical points.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On duality principles and related convex dual formulations suitable for local and global non-convex variational optimization\",\"authors\":\"Fabio Silva Botelho\",\"doi\":\"10.1515/nleng-2022-0343\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract This article develops duality principles, a related convex dual formulation and primal dual formulations suitable for the local and global optimization of non convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg–Landau type equation. Other applications include primal dual variational formulations for a Burger’s type equation and a Navier–Stokes system. We emphasize the novelty here is that the first dual variational formulation developed is convex for a primal formulation which is originally non-convex. Finally, we also highlight the primal dual variational formulations presented have a large region of convexity around any of their critical points.\",\"PeriodicalId\":37863,\"journal\":{\"name\":\"Nonlinear Engineering - Modeling and Application\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Engineering - Modeling and Application\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/nleng-2022-0343\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"ENGINEERING, MECHANICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Engineering - Modeling and Application","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/nleng-2022-0343","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ENGINEERING, MECHANICAL","Score":null,"Total":0}
On duality principles and related convex dual formulations suitable for local and global non-convex variational optimization
Abstract This article develops duality principles, a related convex dual formulation and primal dual formulations suitable for the local and global optimization of non convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg–Landau type equation. Other applications include primal dual variational formulations for a Burger’s type equation and a Navier–Stokes system. We emphasize the novelty here is that the first dual variational formulation developed is convex for a primal formulation which is originally non-convex. Finally, we also highlight the primal dual variational formulations presented have a large region of convexity around any of their critical points.
期刊介绍:
The Journal of Nonlinear Engineering aims to be a platform for sharing original research results in theoretical, experimental, practical, and applied nonlinear phenomena within engineering. It serves as a forum to exchange ideas and applications of nonlinear problems across various engineering disciplines. Articles are considered for publication if they explore nonlinearities in engineering systems, offering realistic mathematical modeling, utilizing nonlinearity for new designs, stabilizing systems, understanding system behavior through nonlinearity, optimizing systems based on nonlinear interactions, and developing algorithms to harness and leverage nonlinear elements.