In a real Hilbert space, we consider two classical problems: the global
minimization of a smooth and convex function $f$ (i.e., a convex optimization
problem) and finding the zeros of a monotone and continuous operator $V$ (i.e.,
a monotone equation). Attached to the optimization problem, first we study the
asymptotic properties of the trajectories generated by a second-order dynamical
system which features a constant viscous friction coefficient and a positive,
monotonically increasing function $b(\cdot)$ multiplying $\nabla f$. For a
generated solution trajectory $y(t)$, we show small $o$ convergence rates
dependent on $b(t)$ for $f(y(t)) – \min f$, and the weak convergence of $y(t)$
towards a global minimizer of $f$. In 2015, Su, Boyd and Cand\’es introduced a
second-order system which could be seen as the continuous-time counterpart of
Nesterov’s accelerated gradient. As the first key point of this paper, we show
that for a special choice for $b(t)$, these two seemingly unrelated dynamical
systems are connected: namely, they are time reparametrizations of each other.
Every statement regarding the continuous-time accelerated gradient system may
be recovered from its Heavy Ball counterpart.
As the second key point of this paper, we observe that this connection
extends beyond the optimization setting. Attached to the monotone equation
involving the operator $V$, we again consider a Heavy Ball-like system which
features an additional correction term which is the time derivative of the
operator along the trajectory. We establish a time reparametrization
equivalence with the Fast OGDA dynamics introduced by Bot, Csetnek and Nguyen
in 2022, which can be seen as an analog of the continuous accelerated gradient
dynamics, but for monotone operators. Again, every statement regarding the Fast
OGDA system may be recovered from a Heavy Ball-like system.

Este artículo explora los viajes en el tiempo y sus implicaciones.

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