Boundedness of trajectories for the generalized Nesterov ODE when r ∈ (1,3)

Determine whether the trajectory X(t) generated by the generalized Nesterov ordinary differential equation ddot{X}(t) + (r/t) dot{X}(t) + ∇f(X(t)) = 0 for t > t0 with r ∈ (1,3), where f: R^n → R is differentiable and convex with at least one minimizer, is bounded for all t ≥ t0 in full generality, without assuming boundedness of the set of minimizers argmin f.

Background

The paper studies point convergence of Nesterov-type dynamics in both continuous and discrete time. For the continuous-time generalized Nesterov ODE ddot{X}(t) + (r/t) dot{X}(t) + ∇f(X(t)) = 0, prior work established convergence in the overdamped regime r > 3. This work proves point convergence in the critical case r = 3 and provides partial results for r ∈ (1,3), including monotonicity of certain energy functions and convergence of f(X(t)) → f*.

For r ∈ (1,3), the authors show boundedness of the trajectory when argmin f is bounded and derive rates on f(X(t)) − f*. However, they cannot establish boundedness in full generality (i.e., without assuming boundedness of argmin f), and explicitly leave this question open.