Second order splitting dynamics with vanishing damping for additively structured monotone inclusions

Second order forward-backward dynamical systems for monotone inclusion problems

We begin by considering second order dynamical systems of the from $\ddot x(t) + γ(t)\dot x(t) + λ(t)B(x(t))=0$, where $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator defined on a real Hilbert space ${\cal H}$, $λ:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function and $γ:[0,+\infty)\rightarrow [0,+\infty)$ a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator $B$. The framework allows to address from similar perspectives…

Solving the additive eigenvalue problem associated to a dynamics of a 2D-traffic system

This is a technical note where we solve the additive eigenvalue problem associated to a dynamics of a 2D-traffic system. The traffic modeling is not explained here. It is available in \cite{Far08}. It consists of a microscopic road traffic model of two circular roads crossing on one junction managed with the priority-to-the-right rule. It is based on Petri nets and minplus algebra. One of our objectives in \cite{Far08} was to derive the fundamental diagram of 2D-traffic, which is the relation between the density and the flow of vehicles. The dynamics of this system, derived from a Petri net de…

Minimax representation of nonexpansive functions and application to zero-sum recursive games

We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on $L^\infty$ spaces and on $\mathbb{R}^n$, which extend results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representati…

Fast optimization via inertial dynamics with closed-loop damping

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…

The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than $1/k^{2}$

The {\it forward-backward algorithm} is a powerful tool for solving optimization problems with a {\it additively separable} and {\it smooth} + {\it nonsmooth} structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov has been proved useful to improve the theoretical rate of convergence for the function values from the standard $\mathcal O(k^{-1})$ down to $\mathcal O(k^{-2})$. In this short paper, we prove that the rate of convergence of a slight variant of Nesterov's accelerated forward-backward method, which produces {\it convergent} sequences, is act…

Tikhonov正则化对单调算子流不仅确保轨迹强收敛，而且加速残差消失

Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals

本研究在实Hilbert空间中证明Tikhonov正则化可确保单调算子流轨迹强收敛至最小范数零点，并加速残差消失，其速率可达O(1/t)，且与Halpern迭代及OGDA动力学相关联。

非线性钻杆系统的分析与 funnel 控制

Analysis and funnel control for nonlinear drill strings

研究非线性钻杆系统输出跟踪问题，通过抽象边界值问题分析模型可解性，提出 novel funnel 控制方法以确保钻头角速度跟踪动态参考轨迹，同时保持跟踪误差在预设性能 funnel 内。

凸优化与单调方程中随机重球动力学的长时间分析

Long-Time Analysis of Stochastic Heavy Ball Dynamics for Convex Optimization and Monotone Equations

该研究在可分实希尔伯特空间中分析带摩擦的随机重球动力学，证明其在凸优化问题中的收敛性与速率，并拓展至含噪单调方程场景，建立对应收敛结果及与二阶动力系统的等价性。

Tikhonov正则化系统的强收敛性与快速收敛率

Strong convergence and fast rates for systems with Tikhonov regularization

研究实希尔伯特空间中带Tikhonov正则化的二阶动力系统，通过消失阻尼和校正项实现轨迹向最小范数解的强收敛及快速收敛率，并将其应用于线性约束凸优化的原始对偶系统。

连续与离散时间下的快速乐观梯度下降上升（OGDA）方法

Fast Optimistic Gradient Descent Ascent (OGDA) method in continuous and discrete time

本文在实希尔伯特空间中提出连续时间二阶动力学及其离散化算法，作为OGDA的加速版本，证明了对单调算子零点的末次迭代快速收敛速率，并数值验证了显式算法的显著优越性。

Momentum-Based Nash Set Seeking over Networks via Multi-Time Scale Hybrid Dynamic Inclusions

Multi-time scale techniques, such as singular perturbations and averaging theory, have played an essential role in the development of distributed Nash equilibrium-seeking algorithms for network systems. Such techniques intrinsically rely on the uniform asymptotic stability properties of the dynamics that evolve in each time scale of the closed-loop system. When such properties are absent, synthesizing multi-time scale Nash equilibrium-seeking algorithms is more challenging and requires additional regularization mechanisms. In this paper, we investigate the synthesis and analysis of these mecha…