Computational Guarantees for Restarted PDHG for LP based on 'Limiting Error Ratios' and LP Sharpness

In recent years, there has been growing interest in solving linear optimization problems - or more simply "LP" - using first-order methods. The restarted primal-dual hybrid gradient method (PDHG) - together with some heuristic techniques - has emerged as a powerful tool for solving huge-scale LPs. However, the theoretical understanding of it and the validation of various heuristic implementation techniques are still very limited. Existing complexity analyses have relied on the Hoffman constant of the LP KKT system, which is known to be overly conservative, difficult to compute (and hence difficult to empirically validate), and fails to offer insight into instance-specific characteristics of the LP problems. These limitations have limited the capability to discern which characteristics of LP instances lead to easy versus difficult LP. With the goal of overcoming these limitations, in this paper we introduce and develop two purely geometry-based condition measures for LP instances: the "limiting error ratio" and the LP sharpness. We provide new computational guarantees for the restarted PDHG based on these two condition measures. For the limiting error ratio, we provide a computable upper bound and show its relationship with the data instance's proximity to infeasibility under perturbation. For the LP sharpness, we prove its equivalence to the stability of the LP optimal solution set under perturbation of the objective function. We validate our computational guarantees in terms of these condition measures via specially constructed instances. Conversely, our computational guarantees validate the practical efficacy of certain heuristic techniques (row preconditioners and step-size tuning) that improve computational performance in practice. Finally, we present computational experiments on LP relaxations from the MIPLIB dataset that demonstrate the promise of various implementation strategies.