不规则非欧几里得整环上的投影动力系统的非线性最优化
Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization
摘要：
连续时间投影动力系统是一类基本的不连续动力系统，其轨迹通过向外投影向量场保持在可行域中。它们在控制系统中的物理饱和建模、运动约束以及研究基于投影的数值优化算法时是必不可少的。基于反馈的连续时间优化方案，依赖于物理系统来执行非线性硬约束的新兴应用的动机，我们研究这些动态的基本性质一般本地欧几里德集。其中，我们提出了使用Krasovskii解决方案，显示其存在性的非凸，不规则子集的低正则性黎曼流形，并调查他们如何与传统的Carathéodory解决方案。此外，我们建立了唯一性的条件，从而引入了一个广义的定义，它是适合于非平坦域的正则性。最后，我们使用这些结果来研究预测梯度流的稳定性和收敛性，作为我们的框架的说明性应用。我们为我们的主要结果提供了简单的反例来说明我们已经很弱的假设的必要性。
英文摘要：
Continuous-time projected dynamical systems are an elementary class of discontinuous dynamical systems with trajectories that remain in a feasible domain by means of projecting outward-pointing vector fields. They are essential when modeling physical saturation in control systems, constraints of motion, as well as studying projection-based numerical optimization algorithms. Motivated by the emerging application of feedback-based continuous-time optimization schemes that rely on the physical system to enforce nonlinear hard constraints, we study the fundamental properties of these dynamics on general locally-Euclidean sets. Among others, we propose the use of Krasovskii solutions, show their existence on nonconvex, irregular subsets of low-regularity Riemannian manifolds, and investigate how they relate to conventional Carathéodory solutions. Furthermore, we establish conditions for uniqueness, thereby introducing a generalized definition of prox-regularity which is suitable for non-flat domains. Finally, we use these results to study the stability and convergence of projected gradient flows as an illustrative application of our framework. We provide simple counter-examples for our main results to illustrate the necessity of our already weak assumptions.
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