Close the logarithmic complexity gap

Determine whether the logarithmic gap between the upper and lower deterministic oracle-complexity bounds for finding an tangent-residual-accurate solution to a composite monotone inclusion problem can be eliminated.

Background

The paper establishes an upper bound of $\widetilde{O}(\varepsilon^{-2/(3p-1)})$ and a matching lower bound of $\Omega(\varepsilon^{-2/(3p-1)})$ for deterministic $p$th-order methods solving composite monotone inclusion problems under the tangent-residual criterion. The remaining discrepancy is logarithmic and arises from the bisection line search used by the Anchored Extra-Proximal Tensor method.

The authors explicitly identify closing this logarithmic discrepancy as an immediate open question, without resolving whether a method can attain the lower bound without the logarithmic overhead.