Improve level-set geometry through preconditioning

Determine whether there exists a linear transformation of the variables, equivalently a change in the inner-product norm, that improves the geometry of the primal-dual level sets for conic linear optimization and thereby improves the conditioning and convergence speed of restarted primal-dual hybrid gradient (rPDHG).

Background

The paper’s convergence guarantees for rPDHG depend on geometric condition measures of the primal-dual duality-gap level sets, especially the ratio of their diameter D_δ to their conic radius r_δ. A smaller ratio corresponds to more favorable geometry and yields a stronger theoretical iteration bound.

The authors therefore identify the unresolved issue of whether a linear variable transformation, or equivalently a change in the underlying inner-product norm, can improve these level-set measures and make a given conic linear optimization instance better conditioned for rPDHG. The paper notes preliminary work using central-path information and heuristic implementations, but leaves the development of effective preconditioning methods unresolved. Further progress could improve both practical performance and theoretical convergence guarantees.

References

Theorem \ref{thm overall complexity clp} begs the question of whether there exists a linear transformation of the variables (or equivalently a change in the inner product norm) that improves the geometry of the level sets $_\delta$ and hence makes the instance better conditioned for rPDHG.

— Level-Set Geometry and the Theoretical Performance of PDHG for Conic Linear Optimization  (2609.19097 - Xiong et al., 16 Sep 2026) in Section 7, item 3, “Summary and research directions”