Extend separation-access optimization to relative-interior geometries

Extend decentralized online upper-linearizable optimization under separation-oracle access to relative-interior geometries such as matroid basis polytopes, thereby enabling applications to additional function classes including one-sided smooth functions.

Background

The Dec-BFTRL guarantee assumes that the feasible domain has a known full-dimensional interior point and supports economical separation-oracle access. Relative-interior domains, including matroid basis polytopes, do not satisfy this full-dimensional-interior assumption in the ambient space. The paper identifies extending the framework to such geometries as an unresolved prerequisite for applying the approach to further objective classes, specifically one-sided smooth functions.

References

Extension to relative-interior geometries such as matroid basis polytopes remains open, and it is necessary before incorporating application to other classes such as the one-sided smooth functions.

— Dec-BFTRL: Squre-Root Regret for Decentralized Online Upper-Linearizable Optimization under Separation Access with Application to Continuous Submodular Maximization  (2608.30271 - Lu et al., 31 Aug 2026) in Section 5, “Conclusion, Limitations, and Future Works”