Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications
<|reference_start|>Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications: In this letter, we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the cost function decreases monotonically along the trajectories of the proximal gradient dynamics. We then introduce a new condition that guarantees exponential convergence of the cost function to its optimal value, and show that this condition implies the proximal Polyak-{\L}ojasiewicz condition. We also show that the proximal Polyak-{\L}ojasiewicz condition guarantees exponential convergence of the cost function. Moreover, we extend these results to time-varying optimization problems, providing bounds for equilibrium tracking. Finally, we discuss applications of these findings, including the LASSO problem, quadratic optimization with polytopic constraints, and certain matrix based problems.<|reference_end|>
@article{gokhale2024proximal,
    title={Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and
  Applications},
    author={Anand Gokhale, Alexander Davydov, Francesco Bullo},
    journal={arXiv preprint arXiv:2409.10664},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.10664},
    primaryClass={math.OC cs.SY eess.SP eess.SY}
}