Adaptive certification for the pure reflected gradient method

Establish whether an adaptive step-size rule can be certified for the pure reflected gradient method, without evaluation-point memory, under monotone Lipschitz variational-inequality assumptions.

Background

The paper’s adaptive convergence certificate relies on the summable evaluation-point filter and its associated telescoping Lyapunov channel. The authors explicitly ask whether that memory can be removed and adaptive convergence can instead be established for the reflected gradient method itself.

References

Whether memory can be removed --- i.e. whether an adaptive step can be certified for PRG itself --- is open; the broken-telescoping mechanism of Section~\ref{sec:adaptive} uses the filter channel nontrivially.

— A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities  (2609.18355 - Shehu, 16 Sep 2026) in Section 6, Discussion and open problems, paragraph “Position among one-call methods”

The full range $(0,(\sqrt2-1)/L)$ for arbitrary nonzero filters and general nonlinear $B$ remains open; its exact performance-estimation formulation is recorded in Appendix~\ref{app:barrier}.

— A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities  (2609.18355 - Shehu, 16 Sep 2026) in Remark 6.3, “the unbounded case: one precise open step”

If \emph{finite identification of the optimal face for PRG on polyhedral sets} holds in general (the only missing ingredient), Theorem~\ref{thm:xaff} upgrades to convergence for all $\lambda|M|<1/\sqrt3$, matching the unconstrained sharp constant.

— A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities  (2609.18355 - Shehu, 16 Sep 2026) in Remark 6.5, “the sharp constant $1/\sqrt3$ via identification”

The unconditional assertion ``$\sum_k|B(u_k)|2<\infty$ for every trajectory at $\lambda<1/(\sqrt3\,L)$'' --- which together with part~(b) would complete the nonlinear sharp-threshold theorem --- is an absolute-stability statement for the feedback loop of Remark~\ref{rem:nonlinear-gap}, whose transfer function has a simple pole on the unit circle; it is open.

— A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities  (2609.18355 - Shehu, 16 Sep 2026) in Theorem 6.10, part (b); Remark 6.13, “connection to absolute stability; the open step”