摘要翻译：
利用梯度动力学研究了凹凸函数的鞍点收敛问题。自从Arrow,Hurwicz和Uzawa在[1]中首次引入这种动力学以来，这种动力学已经广泛地应用于不同的领域，然而，有一些特征使他们的分析变得不平凡。其中包括当考虑的函数不是严格凹凸时缺乏收敛保证，以及次梯度动力学的非光滑性。我们在这两部分的论文中的目的是提供一个明确的刻划一般梯度和次梯度动力学的渐近行为应用于一般凹凸函数。我们证明，尽管这些动力学具有非线性和非光滑性，但它们的$\\omega$-极限集是由轨迹组成的，这些轨迹只求解本文所描述的显式线性问题。更确切地说，在第一部分中，对无约束梯度动力学的渐近行为提供了一个精确的刻划。我们还证明了当不保证收敛到鞍点时，系统的行为可能是有问题的，任意小的噪声导致无界方差。第二部分研究了一类在任意凸域上限制轨迹的一般次梯度动力学，证明了当平衡点存在时，它们的极限轨迹是次梯度动力学在仿射子空间上的解。后者是一类光滑的动力学，具有在第一部分中精确描述的渐近行为，作为显式线性微分方程的解。这些结果被用来制定相应的收敛准则，并在第二部分给出了几个例子和应用。
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英文标题：
《Stability and instability in saddle point dynamics -- Part I》
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作者：
Thomas Holding and Ioannis Lestas
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最新提交年份：
2019
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分类信息：

一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Computer Science        计算机科学
二级分类：Systems and Control        系统与控制
分类描述：cs.SY is an alias for eess.SY. This section includes theoretical and experimental research covering all facets of automatic control systems. The section is focused on methods of control system analysis and design using tools of modeling, simulation and optimization. Specific areas of research include nonlinear, distributed, adaptive, stochastic and robust control in addition to hybrid and discrete event systems. Application areas include automotive and aerospace control systems, network control, biological systems, multiagent and cooperative control, robotics, reinforcement learning, sensor networks, control of cyber-physical and energy-related systems, and control of computing systems.
cs.sy是eess.sy的别名。本部分包括理论和实验研究，涵盖了自动控制系统的各个方面。本节主要介绍利用建模、仿真和优化工具进行控制系统分析和设计的方法。具体研究领域包括非线性、分布式、自适应、随机和鲁棒控制，以及混合和离散事件系统。应用领域包括汽车和航空航天控制系统、网络控制、生物系统、多智能体和协作控制、机器人学、强化学习、传感器网络、信息物理和能源相关系统的控制以及计算系统的控制。
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一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Systems and Control        系统与控制
分类描述：This section includes theoretical and experimental research covering all facets of automatic control systems. The section is focused on methods of control system analysis and design using tools of modeling, simulation and optimization. Specific areas of research include nonlinear, distributed, adaptive, stochastic and robust control in addition to hybrid and discrete event systems. Application areas include automotive and aerospace control systems, network control, biological systems, multiagent and cooperative control, robotics, reinforcement learning, sensor networks, control of cyber-physical and energy-related systems, and control of computing systems.
本部分包括理论和实验研究，涵盖了自动控制系统的各个方面。本节主要介绍利用建模、仿真和优化工具进行控制系统分析和设计的方法。具体研究领域包括非线性、分布式、自适应、随机和鲁棒控制，以及混合和离散事件系统。应用领域包括汽车和航空航天控制系统、网络控制、生物系统、多智能体和协作控制、机器人学、强化学习、传感器网络、信息物理和能源相关系统的控制以及计算系统的控制。
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英文摘要：
  We consider the problem of convergence to a saddle point of a concave-convex function via gradient dynamics. Since first introduced by Arrow, Hurwicz and Uzawa in [1] such dynamics have been extensively used in diverse areas, there are, however, features that render their analysis non trivial. These include the lack of convergence guarantees when the function considered is not strictly concave-convex and also the non-smoothness of subgradient dynamics. Our aim in this two part paper is to provide an explicit characterization to the asymptotic behaviour of general gradient and subgradient dynamics applied to a general concave-convex function. We show that despite the nonlinearity and non-smoothness of these dynamics their $\omega$-limit set is comprised of trajectories that solve only explicit linear ODEs that are characterized within the paper.   More precisely, in Part I an exact characterization is provided to the asymptotic behaviour of unconstrained gradient dynamics. We also show that when convergence to a saddle point is not guaranteed then the system behaviour can be problematic, with arbitrarily small noise leading to an unbounded variance. In Part II we consider a general class of subgradient dynamics that restrict trajectories in an arbitrary convex domain, and show that when an equilibrium point exists their limiting trajectories are solutions of subgradient dynamics on only affine subspaces. The latter is a smooth class of dynamics with an asymptotic behaviour exactly characterized in Part I, as solutions to explicit linear ODEs. These results are used to formulate corresponding convergence criteria and are demonstrated with several examples and applications presented in Part II.
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PDF链接：
https://arxiv.org/pdf/1707.07349