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Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution

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JOURNAL OF DIFFERENTIAL EQUATIONS 2022年 311卷 29-58页

作者： Attouch, Hedy Balhag, Aicha Chbani, Zaki Riahi, Hassan Univ Montpellier CNRS IMAG Montpellier France Univ Bourgogne Franche Comte Inst Math Bourgogne UMR 5584 CNRS F-2100 Dijon France Cadi Ayyad Univ Semlalia Fac Sci Marrakech 40000 Morocco

In a Hilbert space, we provide a fast dynamic approach to the hierarchical minimization problem which consists in finding the minimum norm solution of a convex minimization problem. For this, we study the convergence properties of the trajectories generated by a damped inertial dynamic with Tikhonov regularization. When the time goes to infinity, the Tikhonov regularization parameter is supposed to tend towards zero, not too fast, which is a key property to make the trajectories strongly converge towards the minimizer of minimum norm. According to the structure of the heavy ball method for strongly convex functions, the viscous damping coefficient is proportional to the square root of the Tikhonov regularization parameter. Therefore, it also converges to zero, which will ensure rapid convergence of values. Precisely, under a proper tuning of these parameters, based on Lyapunov's analysis, we show that the trajectories strongly converge towards the minimizer of minimum norm, and we provide the convergence rate of the values. We show a trade off between the property of fast convergence of values, and the property of strong convergence towards the minimum norm solution. This study improves several previous works where this type of results was obtained under restrictive hypotheses. (c) 2021 Elsevier Inc. All rights reserved.

关键词： Accelerated gradient methods Convex optimization Damped inertial dynamics hierarchical minimization Nesterov accelerated gradient method Tikhonov approximation

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Accelerated Gradient Methods Combining Tikhonov Regularization with Geometric Damping Driven by the Hessian

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APPLIED MATHEMATICS AND OPTIMIZATION 2023年第2期88卷 29-29页

作者： Attouch, Hedy Balhag, Aicha Chbani, Zaki Riahi, Hassan Univ Montpellier IMAG CNRS Montpellier France Univ Bourgogne Franche Comte Inst Math Bourgogne UMR 5584 CNRS F-2100 Dijon France Cadi Ayyad Univ Semlalia Fac Sci Marrakech 40000 Morocco

In a Hilbert framework, for general convex differentiable optimization, we consider accelerated gradient dynamics combining Tikhonov regularization with Hessian-driven damping. The temporal discretization of these dynamics leads to a new class of first-order optimization algorithms with favorable properties. The Tikhonov regularization parameter is assumed to tend to zero as time tends to infinity, which preserves equilibria. The presence of the Tikhonov regularization term induces a strong property of convexity which vanishes asymptotically. To take advantage of the fast convergence rates attached to the heavy ball method in the strongly convex case, we consider inertial dynamics where the viscous damping coefficient is proportional to the square root of the Tikhonov regularization parameter, and hence converges to zero. The geometric damping, controlled by the Hessian of the function to be minimized, induces attenuation of the oscillations. Under an appropriate setting of the parameters, based on Lyapunov's analysis, we show that the trajectories provide at the same time several remarkable properties: fast convergence of values, fast convergence of gradients towards zero, and strong convergence to the minimum norm minimizer. We show that the corresponding proximal algorithms share the same properties as continuous dynamics. The numerical illustrations confirm the results obtained. This study extends a previous paper by the authors regarding similar problems without the presence of Hessian driven damping.

关键词： Accelerated gradient methods Convex optimization Damped inertial dynamics Hessian-driven damping hierarchical minimization Nesterov accelerated gradient method Tikhonov approximation

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ASYMPTOTIC BEHAVIOR OF NONAUTONOMOUS MONOTONE AND SUBGRADIENT EVOLUTION EQUATIONS

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2018年第2期370卷 755-790页

作者： Attouch, Hedy Cabot, Alexandre Czarnecki, Marc-Olivier Univ Montpellier 2 Inst Montpellierain Alexander Grothendieck CNRS UMR 5149 Pl Eugene Bataillon F-34095 Montpellier 5 France Univ Bourgogne Franche Comte CNRS UMR 5584 Inst Math Bourgogne F-21000 Dijon France

In a Hilbert setting H, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations (x) over dot (t) + A(t)(x(t)) (sic) 0, where for each t >= 0, A(t) : H paired right arrows H denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory x(.) to a zero of a limit maximal monotone operator A(infinity) as the time variable t tends to + infinity. The crucial point is to use the Brezis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of gphA(infinity) over gphAt tends to zero. This approach gives a sharp and unifying view of this subject. In the case of operators A(t) = partial derivative phi(t) which are subdifferentials of proper closed convex functions phi(t), we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions. Illustrations are given in the field of coupled systems and partial differential equations.

关键词： Nonautonomous monotone inclusion subgradient inclusion multiscale gradient system hierarchical minimization asymptotic behavior Brezis-Haraux function Fitzpatrick function

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Combining fast inertial dynamics for convex optimization with Tikhonov regularization

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2018年第2期457卷 1065-1094页

作者： Attouch, Hedy Chbani, Zaki Riahi, Hassan Univ Montpellier Inst Montpellierain Alexander Grothendieck UMR CNRS 5149 F-34095 Montpellier 5 France Cadi Ayyad Univ Fac Sci Semlalia Math Marrakech 40000 Morocco

In a Hilbert space setting H, we study the convergence properties as t -> +infinity of the trajectories of the second-order differential equation (AVD)(alpha,epsilon) x(t) + alpha/t (x(t) + del Phi(x(t)) + epsilon(t)x(t) = 0, where del Phi is the gradient of a convex continuously differentiable function Phi : H -> R, alpha is a positive parameter, and epsilon(t)x(t) is a Tikhonov regularization term, with epsilon(t) positive, and lim(t) (-> infinity) epsilon(t) = 0. In this damped inertial system, the damping coefficient alpha/t vanishes asymptotically, but not too quickly, a key property to obtain rapid convergence of the values. In the case epsilon(.) equivalent to 0, this dynamic has been highlighted recently by Su, Boyd, and Candes as a continuous version of the Nesterov accelerated gradient method. Depending on the speed of convergence of epsilon(t) to zero, we analyze the convergence properties of the trajectories of (AVD)(alpha,epsilon). We obtain results ranging from the rapid convergence of Phi(x(t)) to min Phi when epsilon(t) decreases rapidly to zero, up to the strong convergence of the trajectories to the element of minimum norm of the set of minimizers of Phi, when epsilon(t) tends slowly to zero. When epsilon(t) = 1/t(r), the critical value of r separating the two above cases is r = 2. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Convex optimization hierarchical minimization Inertial dynamics Nesterov accelerated gradient method Tikhonov approximation Vanishing viscosity

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Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects

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JOURNAL OF DIFFERENTIAL EQUATIONS 2017年第3期262卷 2745-2770页

作者： Attouch, Hedy Czarnecki, Marc-Olivier Univ Montpellier 2 Inst Montpellierain Alexander Grothendieck CNRS UMR 5149 Pl Eugene Bataillon F-34095 Montpellier 5 France

In a Hilbert space H, we study the asymptotic behavior, as time variable t goes to +infinity, of nonautonomous gradient-like dynpmical systems involving inertia and multiscale features. Given Phi : H -> R and Psi : H -> 111 two convex differentiable functions, gamma a positive damping parameter, and epsilon(t) a function oft which tends to zero as t goes to +infinity, we consider the second-order differential equation sic(t) + y(x) over dot (t) + del Phi(x(t)) + epsilon(t)del Psi(x (t)) = 0. This system models the emergence of various collective behaviors in game theory, as well as the asymptotic control of coupled nonlinear oscillators. Assuming that epsilon(t) tends to zero moderately slowly as t goes to infinity, we show that the trajectories converge weakly in H. The limiting equilibria are solutions of the hierarchical minimization problem which consists in minimizing Psi over the set C of minimizers of Phi. As key assumptions, we suppose that integral(+infinity)(0) epsilon(t)dt = +infinity and that, for every p belonging to a convex cone C depending on the data Phi, and Psi integral(+infinity)(0)[Phi*(epsilon(t)p) - sigma(C)(epsilon(t)p)]dt < +infinity where Phi* is the Fenchel conjugate of Phi, and sigma(C) is the support function of C. An application is given to coupled oscillators. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Asymptotic control Convex optimization hierarchical minimization Inertial gradient dynamics Lyapunov analysis Weakly coupled wave equations

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High-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representation

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COMPUTER-AIDED DESIGN 2016年 72卷 52-64页

作者： Ruiz-Girones, Eloi Roca, Xevi Sarrate, Jose UPC Dept Matemat Aplicada MA 3 3 Lab Calcul Numer LaCaN Campus Nord UPC Barcelona 08034 Spain MIT Dept Aeronaut & Astronaut Aerosp Computat Design Lab 77 Massachusetts Ave Cambridge MA 02139 USA

We propose a 3D mesh curving method that converts a straight-sided mesh to an optimal-quality curved high-order mesh that interpolates a CAD boundary representation. The main application of this method is the generation of discrete approximations of curved domains that are valid for simulation analysis with unstructured high-order methods. We devise the method as follows. First, the boundary of a straight sided high-order mesh is curved to match the curves and surfaces of a CAD model. Second, the method minimizes the volume mesh distortion with respect to the coordinates of the inner nodes and the parametric coordinates of the curve and surface nodes. The proposed minimization features untangling capabilities and therefore, it repairs the invalid elements that may arise from the initial curving step. Compared with other mesh curving methods, the only goal of the proposed residual system is to minimize the volume mesh distortion. Furthermore, it is less constrained since the boundary nodes are free to slide on the CAD curves and surfaces. Hence, the proposed method is well suited to generate curved high order meshes of optimal quality from CAD models that contain thin parts or high-curvature entities. To illustrate these capabilities, we generate several curved high-order meshes from CAD models with the implementation detailed in this work. Specifically, we detail a node-by-node non-linear iterative solver that minimizes the proposed objective function in a block Gauss-Seidel manner. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Curved high-order meshing Mesh smoothing Mesh untangling Mesh quality CAD interpolation hierarchical minimization

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The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces

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FIXED POINT THEORY AND APPLICATIONS 2015年第1期2015卷 1-12页

作者： Xu, Hong-Kun Alghamdi, Maryam A. Shahzad, Naseer Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Peoples R China King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia King Abdulaziz Univ Sci Fac Girls Dept Math Jeddah 21491 Saudi Arabia

The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces is established. The strong convergence of this technique is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to variational inequalities, hierarchical minimization problems, and nonlinear evolution equations are included.

关键词： viscosity implicit midpoint rule nonexpansive mapping projection variational inequality hierarchical minimization nonlinear evolution equation

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Coupling the Gradient Method with a General Exterior Penalization Scheme for Convex minimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2012年第1期153卷 123-138页

作者： Peypouquet, Juan Univ Tecn Federico Santa Maria Dept Matemat Valparaiso Chile

In this paper, we propose and analyze an algorithm that couples the gradient method with a general exterior penalization scheme for constrained or hierarchical minimization of convex functions in Hilbert spaces. We prove that a proper but simple choice of the step sizes and penalization parameters guarantees the convergence of the algorithm to solutions for the optimization problem. We also establish robustness and stability results that account for numerical approximation errors, discuss implementation issues and provide examples in finite and infinite dimension.

关键词： Convex optimization hierarchical minimization Exterior penalization Non-autonomous gradient-like systems

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Alternating proximal algorithms with asymptotically vanishing coupling. Application to domain decomposition for PDE's

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OPTIMIZATION 2012年第3期61卷 307-325页

作者： Cabot, A. Frankel, P. Univ Montpellier 2 Dept Math F-34095 Montpellier 5 France

Let X, Y, Z be real Hilbert spaces, let f : X -> R boolean OR {+infinity}, g : Y -> R boolean OR {+infinity} be closed convex functions and let A: X -> Z, B : Y -> Z be linear continuous operators. Given a sequence (gamma(n)) which increases towards infinity as n -> +infinity, we study the following alternating proximal algorithm: (A) x(n+1) = argmin {f(zeta) + 1/2 gamma(n+1) parallel to A zeta - By(n)parallel to(2)(z) + alpha/2 parallel to zeta - x(n)parallel to(2)(x);zeta is an element of x} y(n+1) = argmin {g(eta) + 1/2 gamma(n+1) parallel to Ax(n+1) - B eta parallel to(2)(z) + nu/2 parallel to eta - y(n)parallel to(2)(y);eta is an element of y}, where alpha and nu are positive parameters. If the sequence (gamma(n)) increases moderately slowly towards infinity, the algorithm (A) tends to minimize the function (x, y) bar right arrow parallel to Ax - By parallel to(2)(Z) over the set C = argmin f x argmin g (assumed to be nonempty). An illustration of this result is given in the area of domain decomposition for PDE's.

关键词： convex minimization alternating minimization proximal algorithm hierarchical minimization domain decomposition for PDE's

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HYBRID METHOD FOR DESIGNING EXPLICIT hierarchical FIXED POINT APPROACH TO MONOTONE VARIATIONAL INEQUALITIES

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TAIWANESE JOURNAL OF MATHEMATICS 2012年第4期16卷 1531-1555页

作者： Ceng, Lu-Chuan Lin, Yen-Cherng Petrusel, Adrian Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China Sci Comp Key Lab Shanghai Univ Shanghai 200234 Peoples R China China Med Univ Dept Occupat Safety & Hlth Taichung 404 Taiwan Univ Babes Bolyai Dept Math Cluj Napoca 400084 Romania

Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that F : C -> H is a kappa-Lipschitzian and eta-strongly monotone operator with constants kappa, eta > 0, f : C -> H is L-Lipschitzian with constant L >= 0 and T, V : C -> C are nonexpansive mappings with Fix(T) not equal theta. Let 0 < mu < 2 eta/kappa(2) and 0 <= gamma L < tau, where tau = root 1 - mu(2 eta - mu kappa(2)). Consider the hierarchical monotone variational inequality problem (in short, HMVIP): VI (a): finding z* is an element of Fix(T) such that ((I - V)z*, z - z*) >= 0, for all(z) is an element of Fix(T);VI (b): finding x* is an element of S such that ((mu F - gamma f)x*,x - x*) >= 0, for all(z) is an element of S. Here S denotes the nonempty solution set of the VI (a). This paper combines hybrid steepest-descent method, viscosity method and projection method to design an explicit algorithm, that can be used to find the unique solution of the HMVIP. Strong convergence of the algorithm is proved under very mild conditions. Applications in hierarchical minimization problems are also included.

关键词： Monotone variational inequalities Nonexpansive mapping Iterative algorithm hierarchical fixed point hierarchical minimization Projection

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