A hybrid control framework for fast methods under invexity
Non-Zeno trajectories with exponential rate
A. Sharifi Kolarijani (TU Delft - Team Tamas Keviczky)
P. Mohajerin Esfahani (TU Delft - Team Bart De Schutter)
T. Keviczky (TU Delft - Team Tamas Keviczky)
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Abstract
In this paper, we propose a framework to design a class of fast gradient-based methods in continuous-time that, in comparison with the existing literature including Nesterov's fast-gradient method, features a state-dependent, time-invariant damping term that acts as a feedback control input. The proposed design scheme allows for a user-defined, exponential rate of convergence for a class of nonconvex, unconstrained optimization problems in which the objective function satisfies the so-called Polyak-Łojasiewicz inequality. Formulating the optimization algorithm as a hybrid control system, a state-feedback input is synthesized such that a desired rate of convergence is guaranteed. Furthermore, we establish that the solution trajectories of the hybrid control system are Zeno-free.