"Fast continuous and discrete time methods for monotone equations"Radu Ioan BoţIn this talk, we discuss continuous in time dynamics for the problem of approaching the set of zeros of a single-valued monotone and continuous operator. Such problems are motivated by minimax convex-concave and, in particular, by convex optimization problems with linear equality constraints. We introduce a second-order dynamical system that combines a vanishing damping term with the time derivative of the operator along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. We show that these methods exhibit fast convergence rates for the residual and that the trajectory solution converges to a zero of the underlying operator. For the corresponding implicit and explicit discrete time models with Nesterov's momentum, we prove that they share the asymptotic features of the continuous dynamics. If time permits, we will also discuss the connection between the second-order dynamical system and a Tikhonov regularized first-order dynamical system exhibiting fast and strong convergence rates of the trajectory. |
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