R-linear distance decay under a global error bound

Prove that the distance to the solution set decays R-linearly for the filtered reflected-gradient method under the global error bound \(\dist(x,S)\le c\|x-P_C(x-\lambda B(x))\|\) and a geometrically decaying filter, without assuming strong monotonicity.

Background

The paper establishes strong convergence and a geometrically improving tail bound under the global error bound, but does not derive an R-linear rate for the distance itself. The authors note that the standard residual-contraction argument for Tseng’s method does not directly apply to the reflected update.

References

Under the global error bound eq:eb in place of strong monotonicity, and with a geometrically decaying filter, does $\dist(x_k,S)$ decay $R$-linearly?

eq:eb:

$\dist(x,S)\le c\,\bigl\|x-P_C\bigl(x-\lambda B(x)\bigr)\bigr\| \qquad\forall x\in C,\ \ \lambda=\tfrac1{5L}. $

— A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities  (2609.18355 - Shehu, 16 Sep 2026) in Open Problem 6.1, Section 6, paragraph “Rates”

Whether $\tfrac{3}{10L}$ (or $1/\sqrt3$) is provable for the filtered scheme by a fundamentally different argument --- e.g.\ a non-quadratic Lyapunov function, or a performance-estimation (PEP) certificate --- is, in our view, the most interesting open question raised here; Appendix~\ref{app:barrier} records the barrier computation and the relevant PEP formulation.

— A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities  (2609.18355 - Shehu, 16 Sep 2026) in Remark 6.14, “the constant $1/5$ and the barrier of this framework”; Appendix B, “The barrier of the Lyapunov--Young technique, and a PEP formulation”