检索结果-内蒙古大学图书馆

Asymptotic Equivalence of Evolution Equations Governed by cocoercive operators and Their Forward Discretizations

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第1期182卷 30-48页

作者： Contreras, Andres Peypouquet, Juan Univ Chile Dept Ingn Matemat FCFM Beauchef 851 Santiago Chile Univ Chile FCFM Dept Ingn Matemat Beauchef 851 Santiago Chile Univ Chile FCFM Ctr Modelamiento Matemat CNRSUMI2807 Beauchef 851 Santiago Chile

The purpose of this work is to study discrete approximations of evolution equations governed by cocoercive operators by means of Euler iterations, both in a finite and in an infinite time horizon. On the one hand, we give precise estimations for the distance between iterates of independently generated Euler sequences and use them to obtain bounds for the distance between the state, given by the continuous-time trajectory, and the discrete approximation obtained by the Euler iterations. On the other hand, we establish the asymptotic equivalence between the continuous- and discrete-time systems, under a sharp hypothesis on the step sizes, which can be removed for operators deriving from a potential. As a consequence, we are able to construct a family of smooth functions for which the trajectories/sequences generated by basic first-order methods converge weakly but not strongly, extending the counterexample of Baillon. Finally, we include a few guidelines to address the problem in smooth Banach spaces.

关键词： Asymptotic equivalence cocoercive operators Euler iterations Gradient algorithm

来源：评论

学校读者我要写书评

暂无评论

Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator

引用

OPTIMIZATION 2015年第10期64卷 2223-2252页

作者： Abbas, Boushra Attouch, Hedy Univ Montpellier 2 Inst Math & Modelisat Montpellier UMR CNRS 5149 F-34095 Montpellier 5 France

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators , where is the subdifferential of a convex lower semicontinuous function , and is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials which was considered in Abbas, Attouch, Svaiter JOTA, 2014. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by , and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems, thereby extending the known results. The time discretization of these dynamics gives various forward-backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multicriteria decision processes.

关键词： 47J35 49M37 47J30 90C25 65K15 49M15 34G25 90C53 47J25 forward-backward algorithms structured monotone inclusions cocoercive operators Lyapunov analysis weak asymptotic convergence multiobjective decision dissipative dynamics proximal-gradient method subdifferential operators Levenberg-Marquardt regularization

来源：评论

学校读者我要写书评

暂无评论

ASYMPTOTIC BEHAVIOR OF SECOND-ORDER DISSIPATIVE EVOLUTION EQUATIONS COMBINING POTENTIAL WITH NON-POTENTIAL EFFECTS

引用

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2011年第3期17卷 836-857页

作者： Attouch, Hedy Mainge, Paul-Emile Univ Montpellier 2 CC 51 UMR CNRS 5149 Inst Math & Modelisat Montpellier F-34095 Montpellier 5 France Univ Antilles Guyane DSI CEREGMIA Schoelcher 97233 Martinique France

In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations (u)double over dot(t) + lambda(u)double over dot(t) + del phi(u(t)) + A(u(t)) = 0, where del phi is the gradient operator of a convex differentiable potential function phi : H -> R, A : H -> H is a maximal monotone operator which is assumed to be lambda-cocoercive, and gamma > 0 is a damping parameter. Potential and non-potential effects are associated respectively to del phi and A. Under condition lambda gamma(2) > 1, it is proved that each trajectory asymptotically weakly converges to a zero of del phi + Lambda. This condition, which only involves the non-potential operator and the damping parameter, is sharp and consistent with time rescaling. Passing from weak to strong convergence of the trajectories is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.

关键词： Second-order evolution equations asymptotic behavior dissipative systems maximal monotone operators potential and non-potential operators cocoercive operators Tikhonov regularization heavy ball with friction dynamical system constrained optimization coupled systems dynamical games Nash equilibria

来源：评论

学校读者我要写书评

暂无评论

COUPLING FORWARD-BACKWARD WITH PENALTY SCHEMES AND PARALLEL SPLITTING FOR CONSTRAINED VARIATIONAL INEQUALITIES

引用

SIAM JOURNAL ON OPTIMIZATION 2011年第4期21卷 1251-1274页

作者： Attouch, Hedy Czarnecki, Marc-Olivier Peypouquet, Juan Univ Montpellier 2 CNRS UMR 5149 Inst Math & Modelisat Montpellier F-34095 Montpellier 5 France Univ Tecn Federico Santa Maria Dept Matemat Valparaiso Chile

We are concerned with the study of a class of forward-backward penalty schemes for solving variational inequalities 0 is an element of Ax + N-C(x) where H is a real Hilbert space, A : H paired right arrows H is a maximal monotone operator, and N-C is the outward normal cone to a closed convex set C subset of H. Let Psi : H -> R be a convex differentiable function whose gradient is Lipschitz continuous and which acts as a penalization function with respect to the constraint x is an element of C. Given a sequence (beta(n)) of penalization parameters which tends to infinity, and a sequence of positive time steps (lambda(n)) is an element of l(2)\l(1), we consider the diagonal forward-backward algorithm x(n+1) = (I + lambda(n)A)(-1)(x(n) - lambda(n)beta(n)del Psi(x(n))). Assuming that (beta(n)) satisfies the growth condition limsup(n ->infinity)lambda(n)beta(n) < 2/theta (where theta is the Lipschitz constant of del Psi), we obtain weak ergodic convergence of the sequence (x(n)) to an equilibrium for a general maximal monotone operator A. We also obtain weak convergence of the whole sequence (x(n)) when A is the subdifferential of a proper lower-semicontinuous convex function. As a key ingredient of our analysis, we use the cocoerciveness of the operator del Psi. When specializing our results to coupled systems, we bring new light to Passty's theorem and obtain convergence results of new parallel splitting algorithms for variational inequalities involving coupling in the constraint. We also establish robustness and stability results that account for numerical approximation errors. An illustration of compressive sensing is given.

关键词： cocoercive operators compressive sensing constrained convex optimization coupled systems forward-backward algorithms hierarchical optimization maximal monotone operators parallel splitting methods Passty's theorem penalization methods variational inequalities

来源：评论

学校读者我要写书评

暂无评论

Convergence of a Relaxed Inertial Forward-Backward Algorithm for Structured Monotone Inclusions

引用

APPLIED MATHEMATICS AND OPTIMIZATION 2019年第3期80卷 547-598页

作者： Attouch, Hedy Cabot, Alexandre Univ Montpellier CNRS Inst Montpellierain Alexander Grothendieck UMR 5149 Pl Eugene Bataillon F-34095 Montpellier 5 France Univ Bourgogne Franche Comte CNRS Inst Math Bourgogne UMR 5584 F-21000 Dijon France

In a Hilbert space H, we study the convergence properties of a class of relaxed inertial forward-backward algorithms. They aim to solve structured monotone inclusions of the form Ax + Bx (sic) 0 where A : H -> 2(H) is a maximally monotone operator and B : H -> H is a cocoercive operator. We extend to this class of problems the acceleration techniques initially introduced by Nesterov, then developed by Beck and Teboulle in the case of structured convex minimization (FISTA). As an important element of our approach, we develop an inertial and parametric version of the Krasnoselskii-Mann theorem, where joint adjustment of the inertia and relaxation parameters plays a central role. This study comes as a natural extension of the techniques introduced by the authors for the study of relaxed inertial proximal algorithms. An illustration is given to the inertial Nash equilibration of a game combining non-cooperative and cooperative aspects.

关键词： Structured monotone inclusions Inertial forward-backward algorithms cocoercive operators Relaxation Convergence rate Inertial Krasnoselskii-Mann iteration Nash equilibration

来源：评论

学校读者我要写书评

暂无评论

Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms

引用

FIXED POINT THEORY AND ALGORITHMS FOR SCIENCES AND ENGINEERING 2021年第1期2021卷 1-30页

作者： Adly, Samir Attouch, Hedy Vo, Van Nam Univ Limoges Lab XLIM 123 Ave Albert Thomas F-87060 Limoges France Univ Montpellier IMAG CNRS Pl Eugene Bataillon F-34095 Montpellier 5 France

In a Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator A=del f+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A= \nabla f +B $\end{document}, where del f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as t ->+infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\to +\infty $\end{document} of the generated trajectories towards the zeros of del f+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla f +B$\end{document}. The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate t

关键词： Proximal-gradient algorithms Inertial methods Hessian driven damping Nonpotential term cocoercive operators

来源：评论

学校读者我要写书评

暂无评论

Fast Continuous Dynamics Inside the Graph of Maximally Monotone operators

引用

SET-VALUED AND VARIATIONAL ANALYSIS 2023年第1期31卷 5-5页

作者： Mainge, Paul-Emile Weng-Law, Andre Univ Antilles FWI MEMIAD Campus Schoelcher F-97233 Pointe a Pitre Martinique France

In a Hilbert framework, we introduce a new class of fast continuous dissipative dynamical systems for approximating zeroes of an arbitrary maximally monotone operator. This system originates from some change of variable operated in a continuous Nesterov-like model that is driven by the Yosida regularization of the operator and that involves an asymptotic vanishing damping. The proposed model (based upon the Minty representation of maximally monotone operators) is investigated through a first-order reformulation that amounts to dynamics involving three variables: a couple of two variables that lies in the graph of the considered operator, along with an auxiliary variable. We establish the weak convergence towards zeroes of the operator for two of the trajectories, as well as fast convergence to zero of their velocities. We also prove a fast convergence property to zero for the third trajectory and for its velocity. It turns out that our model offers a new and well-adapted framework for discrete counterparts. Several algorithms are then suggested relative to monotone inclusions. Some of them recover some recently developed inertial proximal algorithms that incorporate a correction term in addition to the classical momentum term.

关键词： Asymptotic behavior First-order differential equation Dissipative systems Maximal monotone operators Minty's representation cocoercive operators Yosida approximation Nesterov-like acceleration

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：