Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization
Abstract
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 Carath\'eodory 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.
Cite
@article{arxiv.1809.04831,
title = {Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization},
author = {Adrian Hauswirth and Saverio Bolognani and Florian Dörfler},
journal= {arXiv preprint arXiv:1809.04831},
year = {2020}
}