Abstract:

In this paper we propose a new family of continuous-time optimization algorithms based on discontinuous second order gradient optimization flows, with finite-time convergence guarantees to local optima, for locally strongly convex (time-varying) cost functions. To analyze our flows, we first extend a well-know Lyapunov inequality condition for finite-time stability, to the case of (time-varying) differential inclusions. We then prove the convergence of these second-order flows in finite-time. In some particular cases, we can show that the finite-time convergence can be pre-defined by the user. We propose a robustification of the flows to bounded additive uncertainties, and extend some of the results to the case of constrained optimization. We show the performance of these flows on well-know optimization benchmarks, namely, the Rosenbrock function, and the Rastringin function.