Sharper Merit-Function Correction Coefficients for Max-M Non-Monotone Direct Search

Determine whether sharper correction coefficients than those used in the merit function for deterministic probabilistic-descent max-$M$ non-monotone direct search can be constructed, thereby improving the geometric dependence of the iteration-complexity bound on the memory length $M$.

Background

For the deterministic probabilistic-descent algorithm, the paper introduces a merit function with ordered squared-stepsize corrections. The resulting decrease constant decreases geometrically with the memory length MM, causing the expected iteration bound to contain a factor of order [?][?]gamma{2M}.</sup></p><p>Theauthorsexplicitlynotethatthisgeometricdependencearisesfromtheparticularfeasiblecorrectioncoefficientsselectedintheiranalysisratherthanfromaprovenlowerboundapplyingtoeverymax−.</sup></p> <p>The authors explicitly note that this geometric dependence arises from the particular feasible correction coefficients selected in their analysis rather than from a proven lower bound applying to every max-Mmethod.Theunresolvedproblemisthereforetodeterminewhetheralternative,sharpercoefficientscanyieldimproveddependenceon method. The unresolved problem is therefore to determine whether alternative, sharper coefficients can yield improved dependence on M$ while preserving the required merit-function decrease and complexity guarantees.

References

The explicit constant $\nu$ in Lemma~\ref{l3.2} decreases geometrically with~$M$, so the resulting bound contains a factor of order $\gamma{2M}$. In contrast, the complete-poll analysis gives linear dependence on~$M$. The geometric factor is a consequence of the particular feasible correction coefficients used here, not a lower bound for every max-$M$ method; hence, the possibility of sharper coefficients is open.

— Non-monotone direct-search methods for deterministic and stochastic derivative-free optimization  (2609.11567 - Ding et al., 10 Sep 2026) in Section 3, subsection “Max-$M$ non-monotone direct search with probabilistic descent,” immediately before Section 4