{"title":"由吉洪诺夫项正则化的重球法。值和轨迹的同时收敛","authors":"Akram Chahid Bagy, Z. Chbani, H. Riahi","doi":"10.3934/eect.2022046","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">\\begin{document}$ f: {\\mathcal H} \\rightarrow \\mathbb{R} $\\end{document}</tex-math></inline-formula> be a convex differentiable function whose solution set <inline-formula><tex-math id=\"M2\">\\begin{document}$ {{\\rm{argmin}}}\\; f $\\end{document}</tex-math></inline-formula> is nonempty. To attain a solution of the problem <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\min_{\\mathcal H}f $\\end{document}</tex-math></inline-formula>, we consider the second order dynamic system <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\;\\ddot{x}(t) + \\alpha \\, \\dot{x}(t) + \\beta (t) \\nabla f(x(t)) + c x(t) = 0 $\\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\beta $\\end{document}</tex-math></inline-formula> is a positive function such that <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\lim_{t\\rightarrow +\\infty}\\beta(t) = +\\infty $\\end{document}</tex-math></inline-formula>. By imposing adequate hypothesis on first and second order derivatives of <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\beta $\\end{document}</tex-math></inline-formula>, we simultaneously prove that the value of the objective function in a generated trajectory converges in order <inline-formula><tex-math id=\"M8\">\\begin{document}$ {\\mathcal O}\\big(\\frac{1}{\\beta(t)}\\big) $\\end{document}</tex-math></inline-formula> to the global minimum of the objective function, that the trajectory strongly converges to the minimum norm element of <inline-formula><tex-math id=\"M9\">\\begin{document}$ {{\\rm{argmin}}}\\; f $\\end{document}</tex-math></inline-formula> and that <inline-formula><tex-math id=\"M10\">\\begin{document}$ \\Vert \\dot{x}(t)\\Vert $\\end{document}</tex-math></inline-formula> converges to zero in order <inline-formula><tex-math id=\"M11\">\\begin{document}$ \\mathcal{O} \\big( \\sqrt{\\frac{\\dot{\\beta}(t)}{\\beta (t)}}+ e^{-\\mu t} \\big) $\\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id=\"M12\">\\begin{document}$ \\mu<\\frac{\\alpha}2 $\\end{document}</tex-math></inline-formula>. We then present two choices of <inline-formula><tex-math id=\"M13\">\\begin{document}$ \\beta $\\end{document}</tex-math></inline-formula> to illustrate these results. On the basis of the Moreau regularization technique, we extend these results to non-smooth convex functions with extended real values.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"6 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The Heavy ball method regularized by Tikhonov term. Simultaneous convergence of values and trajectories\",\"authors\":\"Akram Chahid Bagy, Z. Chbani, H. Riahi\",\"doi\":\"10.3934/eect.2022046\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\\\"M1\\\">\\\\begin{document}$ f: {\\\\mathcal H} \\\\rightarrow \\\\mathbb{R} $\\\\end{document}</tex-math></inline-formula> be a convex differentiable function whose solution set <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ {{\\\\rm{argmin}}}\\\\; f $\\\\end{document}</tex-math></inline-formula> is nonempty. To attain a solution of the problem <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ \\\\min_{\\\\mathcal H}f $\\\\end{document}</tex-math></inline-formula>, we consider the second order dynamic system <inline-formula><tex-math id=\\\"M4\\\">\\\\begin{document}$ \\\\;\\\\ddot{x}(t) + \\\\alpha \\\\, \\\\dot{x}(t) + \\\\beta (t) \\\\nabla f(x(t)) + c x(t) = 0 $\\\\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ \\\\beta $\\\\end{document}</tex-math></inline-formula> is a positive function such that <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ \\\\lim_{t\\\\rightarrow +\\\\infty}\\\\beta(t) = +\\\\infty $\\\\end{document}</tex-math></inline-formula>. By imposing adequate hypothesis on first and second order derivatives of <inline-formula><tex-math id=\\\"M7\\\">\\\\begin{document}$ \\\\beta $\\\\end{document}</tex-math></inline-formula>, we simultaneously prove that the value of the objective function in a generated trajectory converges in order <inline-formula><tex-math id=\\\"M8\\\">\\\\begin{document}$ {\\\\mathcal O}\\\\big(\\\\frac{1}{\\\\beta(t)}\\\\big) $\\\\end{document}</tex-math></inline-formula> to the global minimum of the objective function, that the trajectory strongly converges to the minimum norm element of <inline-formula><tex-math id=\\\"M9\\\">\\\\begin{document}$ {{\\\\rm{argmin}}}\\\\; f $\\\\end{document}</tex-math></inline-formula> and that <inline-formula><tex-math id=\\\"M10\\\">\\\\begin{document}$ \\\\Vert \\\\dot{x}(t)\\\\Vert $\\\\end{document}</tex-math></inline-formula> converges to zero in order <inline-formula><tex-math id=\\\"M11\\\">\\\\begin{document}$ \\\\mathcal{O} \\\\big( \\\\sqrt{\\\\frac{\\\\dot{\\\\beta}(t)}{\\\\beta (t)}}+ e^{-\\\\mu t} \\\\big) $\\\\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id=\\\"M12\\\">\\\\begin{document}$ \\\\mu<\\\\frac{\\\\alpha}2 $\\\\end{document}</tex-math></inline-formula>. We then present two choices of <inline-formula><tex-math id=\\\"M13\\\">\\\\begin{document}$ \\\\beta $\\\\end{document}</tex-math></inline-formula> to illustrate these results. On the basis of the Moreau regularization technique, we extend these results to non-smooth convex functions with extended real values.</p>\",\"PeriodicalId\":48833,\"journal\":{\"name\":\"Evolution Equations and Control Theory\",\"volume\":\"6 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Evolution Equations and Control Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3934/eect.2022046\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Evolution Equations and Control Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/eect.2022046","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The Heavy ball method regularized by Tikhonov term. Simultaneous convergence of values and trajectories
Let \begin{document}$ f: {\mathcal H} \rightarrow \mathbb{R} $\end{document} be a convex differentiable function whose solution set \begin{document}$ {{\rm{argmin}}}\; f $\end{document} is nonempty. To attain a solution of the problem \begin{document}$ \min_{\mathcal H}f $\end{document}, we consider the second order dynamic system \begin{document}$ \;\ddot{x}(t) + \alpha \, \dot{x}(t) + \beta (t) \nabla f(x(t)) + c x(t) = 0 $\end{document}, where \begin{document}$ \beta $\end{document} is a positive function such that \begin{document}$ \lim_{t\rightarrow +\infty}\beta(t) = +\infty $\end{document}. By imposing adequate hypothesis on first and second order derivatives of \begin{document}$ \beta $\end{document}, we simultaneously prove that the value of the objective function in a generated trajectory converges in order \begin{document}$ {\mathcal O}\big(\frac{1}{\beta(t)}\big) $\end{document} to the global minimum of the objective function, that the trajectory strongly converges to the minimum norm element of \begin{document}$ {{\rm{argmin}}}\; f $\end{document} and that \begin{document}$ \Vert \dot{x}(t)\Vert $\end{document} converges to zero in order \begin{document}$ \mathcal{O} \big( \sqrt{\frac{\dot{\beta}(t)}{\beta (t)}}+ e^{-\mu t} \big) $\end{document} where \begin{document}$ \mu<\frac{\alpha}2 $\end{document}. We then present two choices of \begin{document}$ \beta $\end{document} to illustrate these results. On the basis of the Moreau regularization technique, we extend these results to non-smooth convex functions with extended real values.
期刊介绍:
EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE''s and FDEs. Topics include:
* Modeling of physical systems as infinite-dimensional processes
* Direct problems such as existence, regularity and well-posedness
* Stability, long-time behavior and associated dynamical attractors
* Indirect problems such as exact controllability, reachability theory and inverse problems
* Optimization - including shape optimization - optimal control, game theory and calculus of variations
* Well-posedness, stability and control of coupled systems with an interface. Free boundary problems and problems with moving interface(s)
* Applications of the theory to physics, chemistry, engineering, economics, medicine and biology