In this paper we propose new numerical algorithms in the setting of unconstrained optimization problems and we study the rate of convergence in the iterates of the objective function. Furthermore, our algorithms are based upon splitting and symplectic methods and they preserve the energy properties of the inherent continuous dynamical system that contains a Hessian perturbation. At the same time, we show that Nesterov gradient method is equivalent to a Lie-Trotter splitting applied to a Hessian driven damping system. Finally, some numerical experiments are presented in order to validate the theoretical results.
In a Hilbert setting, we develop fast methods for convex unconstrained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three coefficients varying with time, one is a viscous damping coefficient, the second is attached to the Hessian-driven damping, the third is a time scaling coefficient. We study the convergence rate of the values under general conditions involving the damping and the time scale coefficients. The obtained results a…
In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form $\nabla^2 f (x(t)) \dot{x} (t)$. By treating this term as the time derivative of $ \nabla f (x (t)) $, this gives, in discretized form, first-order algorithms in time and space. In addition to the convergence properties attached to Nesterov-…
We first study the fast minimization properties of the trajectories of the second-order evolution equation $$\ddot{x}(t) + \fracα{t} \dot{x}(t) + β\nabla^2 Φ(x(t))\dot{x} (t) + \nabla Φ(x(t)) = 0,$$ where $Φ:\mathcal H\to\mathbb R$ is a smooth convex function acting on a real Hilbert space $\mathcal H$, and $α$, $β$ are positive parameters. This inertial system combines an isotropic viscous damping which vanishes asymptotically, and a geometrical Hessian driven damping, which makes it naturally related to Newton's and Levenberg-Marquardt methods. For $α\geq 3$, $β>0$, along any trajectory, fas…
Fast convergence rates and trajectory convergence of a Tikhonov regularized inertial primal\mbox{-}dual dynamical system with time scaling and vanishing damping
In this paper, we propose a second-order continuous primal-dual dynamical system with time-dependent positive damping terms for a separable convex optimization problem with linear equality constraints. By the Lyapunov function approach, we investigate asymptotic properties of the proposed dynamical system as the time $t\to+\infty$. The convergence rates are derived for different choices of the damping coefficients. We also show that the obtained results are robust under external perturbations.
First-order optimization algorithms can be considered as a discretization of ordinary differential equations (ODEs) \cite{su2014differential}. In this perspective, studying the properties of the corresponding trajectories may lead to convergence results which can be transfered to the numerical scheme. In this paper we analyse the following ODE introduced by Attouch et al. in \cite{attouch2016fast}: \begin{equation*} \forall t\geqslant t_0,~\ddot{x}(t)+\fracα{t}\dot{x}(t)+βH_F(x(t))\dot{x}(t)+\nabla F(x(t))=0,\end{equation*} where $α>0$, $β>0$ and $H_F$ denotes the Hessian of $F$. This ODE can …
In a Hilbert space $H$, in order to develop fast optimization methods, we analyze the asymptotic behavior, as time $t$ tends to infinity, of inertial continuous dynamics where the damping acts as a closed-loop control. The function $f: H \to R$ to be minimized (not necessarily convex) enters the dynamic through it gradient, which is assumed to be Lipschitz continuous on the bounded subsets of $H$. This gives autonomous dynamical systems with nonlinear damping and nonlinear driving force. We first consider the case where the damping term $\partial ϕ(\dot{x}(t))$ acts as a closed-loop control of…
In Hilbert space, we propose a family of primal-dual dynamical system for affine constrained convex optimization problem. Several damping coefficients, time scaling coefficients, and perturbation terms are thus considered. By constructing the energy functions, we investigate the convergence rates with different choices of the damping coefficients and time scaling coefficients. Our results extend the inertial dynamical approaches for unconstrained convex optimization problems to affine constrained convex optimization problems.
We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, whose trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plai…
In a Hilbert space setting, for convex optimization, we show the convergence of the iterates to optimal solutions for a class of accelerated first-order algorithms. They can be interpreted as discrete temporal versions of an inertial dynamic involving both viscous damping and Hessian-driven damping. The asymptotically vanishing viscous damping is linked to the accelerated gradient method of Nesterov while the Hessian driven damping makes it possible to significantly attenuate the oscillations. By treating the Hessian-driven damping as the time derivative of the gradient term, this gives, in di…
In this article a family of second order ODEs associated to inertial gradient descend is studied. These ODEs are widely used to build trajectories converging to a minimizer $x^*$ of a function $F$, possibly convex. This family includes the continuous version of the Nesterov inertial scheme and the continuous heavy ball method. Several damping parameters, not necessarily vanishing, and a perturbation term $g$ are thus considered. The damping parameter is linked to the inertia of the associated inertial scheme and the perturbation term $g$ is linked to the error that can be done on the gradient …
In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are respectively associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provide…
We study the behavior of the trajectories of a second-order differential equation with vanishing damping, governed by the Yosida regularization of a maximally monotone operator with time-varying index, along with a new {\em Regularized Inertial Proximal Algorithm} obtained by means of a convenient finite-difference discretization. These systems are the counterpart to accelerated forward-backward algorithms in the context of maximally monotone operators. A proper tuning of the parameters allows us to prove the weak convergence of the trajectories to zeroes of the operator. Moreover, it is possi…
In a Hilbert setting, for convex differentiable optimization, we consider accelerated gradient dynamics combining Tikhonov regularization with Hessian-driven damping. 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 convexity property which vanishes asymptotically. To take advantage of the exponential convergence rates attached to the heavy ball method in the strongly convex case, we consider the inertial dynamic where the viscous damping coefficient is taken…
In a real Hilbert space $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order evolution equation $$ \ddot{x}(t) + \fracα{t} \dot{x}(t) + \nabla Φ(x(t)) = 0, $$ where $\nabla Φ$ is the gradient of a convex continuously differentiable function $Φ: \mathcal H \rightarrow \mathbb R$, and $α$ is a positive parameter. In this inertial system, the viscous damping coefficient $\fracα{t}$ vanishes asymptotically in a moderate way. For $α> 3$, we show that any trajectory converges weakly to a minimizer of $Φ$, just assuming that the set of min…
In a Hilbertian framework, for the minimization of a general convex differentiable function $f$, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of $f$ with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time $t$, is associated with the strongly convex function obtained by adding to $f$ a Tikhonov regularization term with vanishing coefficient $ε(t)$. In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regulariza…
In a Hilbert space setting $\mathcal H$, we study the convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \mbox{(AVD)}_{α, ε} \quad \quad \ddot{x}(t) + \fracα{t} \dot{x}(t) + \nabla Φ(x(t)) + ε(t) x(t) =0, \end{equation*} where $\nablaΦ$ is the gradient of a convex continuously differentiable function $Φ: \mathcal H \to \mathbb R$, $α$ is a positive parameter, and $ε(t) x(t)$ is a Tikhonov regularization term, with $\lim_{t \to \infty}ε(t) =0$. In this damped inertial system, the damping coefficient $\fracα{t}$ vanishes asy…
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 $f$ of minimum norm. According to the structure of the heavy ball method …
University of Vienna（维也纳大学）；University of Southampton（南安普顿大学）
We introduce a relaxed inertial forward-backward-forward (RIFBF) splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitz continuous operator. This work aims to extend Tseng's forward-backward-forward method by both using inertial effects as well as relaxation parameters. We formulate first a second order dynamical system which approaches the solution set of the monotone inclusion problem to be solved and provide an asymptotic analysis for its trajectories. We provide for RIFBF, which follows by explicit time di…
In a Hilbert space setting $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation $ \ddot{x}(t) + \fracα{t} \dot{x}(t) + \nabla Φ(x(t)) = g(t)$, where $\nablaΦ$ is the gradient of a convex continuously differentiable function $Φ: \mathcal H \to \mathbb R$, $α$ is a positive parameter, and $g: [t_0, + \infty[ \rightarrow \mathcal H$ is a "small" perturbation term. In this damped inertial system, the viscous damping coefficient $\fracα{t}$ vanishes asymptotically, but not too rapidly. For $α\geq 3$, and $\int_{t_0}…
Ensemble Kalman Inversion as an Inertial Interacting Particle System
Institute for Geometry and Applied Mathematics, RWTH Aachen University（亚琛工业大学几何与应用数学研究所）；Department of Mathematics and Applied Mathematics, University of Pretoria（比勒陀利亚大学数学与应用数学系）；Department of Mathematics, Sapienza University of Rome（罗马第一大学数学系）
In this manuscript, we study the properties of a family of second-order differential equations with damping, its discretizations and their connections with accelerated optimization algorithms for $m$-strongly convex and $L$-smooth functions. In particular, using the Linear Matrix Inequality LMI framework developed by \emph{Fazlyab et. al. $(2018)$}, we derive analytically a (discrete) Lyapunov function for a two-parameter family of Nesterov optimization methods, which allows for the complete characterization of their convergence rate. In the appropriate limit, this family of methods may be see…
