  While many approaches were developed for obtaining worst-case complexity
bounds for first-order optimization methods in the last years, there remain
theoretical gaps in cases where no such bound can be found. In such cases, it
is often unclear whether no such bound exists (e.g., because the algorithm
might fail to systematically converge) or simply if the current techniques do
not allow finding them.
  In this work, we propose an approach to automate the search for cyclic
trajectories generated by first-order methods. This provides a constructive
approach to show that no appropriate complexity bound exists, thereby
complementing the approaches providing sufficient conditions for convergence.
Using this tool, we provide ranges of parameters for which some of the famous
heavy-ball, Nesterov accelerated gradient, inexact gradient descent, and
three-operator splitting algorithms fail to systematically converge, and show
that it nicely complements existing tools searching for Lyapunov functions.
