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Fast Gradient-Based Methods with Exponential Rate: A Hybrid Control Framework
Arman Sharifi Kolarijani · Peyman Mohajerin Esfahani · Tamas Keviczky

Fri Jul 13 09:15 AM -- 12:00 PM (PDT) @ Hall B #17
Ordinary differential equations, and in general a dynamical system viewpoint, have seen a resurgence of interest in developing fast optimization methods, mainly thanks to the availability of well-established analysis tools. In this study, we pursue a similar objective and propose a class of hybrid control systems that adopts a 2nd-order differential equation as its continuous flow. A distinctive feature of the proposed differential equation in comparison with the existing literature is a state-dependent, time-invariant damping term that acts as a feedback control input. Given a user-defined scalar $\alpha$, it is shown that the proposed control input steers the state trajectories to the global optimizer of a desired objective function with a guaranteed rate of convergence $\mathcal{O}(e^{-\alpha t})$. Our framework requires that the objective function satisfies the so called Polyak--{\L}ojasiewicz inequality. Furthermore, a discretization method is introduced such that the resulting discrete dynamical system possesses an exponential rate of convergence.

Author Information

Arman Sharifi Kolarijani (Delft University of Technology)
Peyman Mohajerin Esfahani (Delft University of Technology)
Tamas Keviczky (Delft University of Technology)

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