Convergence guarantee for adaptive PDLP step-size rules

Establish a convergence guarantee for the adaptive step-size acceptance rule used in practical PDLP-style solvers, which accepts or reduces trial step sizes based on the current update.

Background

The paper reviews adaptive step-size rules used in PDLP, cuPDLP.jl/C, MPAX, and related solvers. These methods test a trial update and reduce the step size when an inequality involving the current primal and dual displacements fails. The test is motivated by the nonexpansiveness analysis of the PDHG operator, but it can permit steps beyond the sufficient spectral-norm bound and may require rejected trial updates, which incur additional matrix–vector products.

The paper explicitly notes that a convergence guarantee for this practical adaptive rule has not been established. Consequently, a rigorous analysis is needed to determine whether the rule guarantees convergence, and under what assumptions on the LP, parameter updates, and sequence of accepted trial steps.

References

Although the test is motivated by the nonexpansiveness proof of the PDHG map, to the best of our knowledge, no convergence guarantee has been established for the adaptive step-size rule used in practical PDLP-style solvers~\citep{applegate2021practical,lu2024halpern}.

— Reinforcement Learning to Accelerate Primal-Dual Hybrid Gradient for Linear Programming  (2610.01546 - Sul et al., 1 Oct 2026) in Appendix, Section 2, subsection “Step size” (Appendix \ref{app:pdlp-step-size})