Accelerated objective-value rate at the critical endpoint a equals one-half

Establish an accelerated objective-value estimate for the endogenous gradient-memory accelerated gradient flow at the critical parameter value a=1/2, possibly under additional assumptions on the objective function or by constructing a different Lyapunov functional.

Background

For 1/2<a<1, the paper derives an accelerated objective-value estimate of order O(t{-2}) from a Lyapunov functional containing the term t2(F(x(t))-F*). At the endpoint a=1/2, the coefficient multiplying this objective-value term vanishes, so the available Lyapunov functional controls only the accelerated phase variable and does not directly provide a value-rate estimate.

The authors explicitly identify the critical endpoint as unresolved and indicate two possible routes: imposing additional assumptions on F or developing a different Lyapunov functional that can recover an objective-value convergence rate.

References

When $a=\frac12,$ the coefficient $2a-1$ vanishes. The Lyapunov functional $E_{\frac12}(t)=\frac12|y(t)|2$ controls only the phase variable and does not contain the objective-value term $t2\bigl(F(x(t))-F\star\bigr).$ Therefore the preceding argument does not yield an accelerated objective-value estimate at the endpoint $a=\frac12$. Additional assumptions or a different Lyapunov functional may be required to obtain a value-rate statement in this critical regime. This critical case remains open.

— Accelerated Gradient Flow with Endogenous Gradient-Memory Anchor: Selection and Restoring Damping  (2608.25312 - Izuchukwu et al., 26 Aug 2026) in Remark 'The endpoint a=1/2', Section 5.2, subsection 'Accelerated Objective-value estimates'

Several questions remain open. It would be natural to determine whether the anchor-induced selection principle extends beyond affine solution sets. Another direction is to investigate whether stronger assumptions on $F$, such as error-bound conditions , Polyak--Lojasiewicz conditions , or Kurdyka--{\L}ojasiewicz conditions , lead to sharper convergence rates for the distance to the solution set, the gradient norm, or the trajectory .

— Accelerated Gradient Flow with Endogenous Gradient-Memory Anchor: Selection and Restoring Damping  (2608.25312 - Izuchukwu et al., 26 Aug 2026) in Section 7, Conclusion