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There has been a recent surge in development of first-order methods (FOMs)for solving huge-scale linear programming (LP) problems. The attractiveness ofFOMs for LP stems in part from the fact that they avoid costly matrixfactorization computation. However, the efficiency of FOMs is significantlyinfluenced - both in theory and in practice - by certain instance-specific LPcondition measures. Xiong and Freund recently showed that the performance ofthe restarted primal-dual hybrid gradient method (PDHG) is predominantlydetermined by two specific condition measures: LP sharpness and Limiting ErrorRatio. In this paper we examine the relationship between these two measures,particularly in the case when the optimal solution is unique (which is generic- at least in theory), and we present an upper bound on the Limiting ErrorRatio involving the reciprocal of the LP sharpness. This shows that in LPinstances where there is a dual nondegenerate optimal solution, thecomputational complexity of restarted PDHG can be characterized solely in termsof LP sharpness and the distance to optimal solutions, and simplifies thetheoretical complexity upper bound of restarted PDHG for these instances.

On the Relation Between LP Sharpness and Limiting Error Ratio and Complexity Implications for Restarted PDHG