On the Relation Between LP Sharpness and Limiting Error Ratio and Complexity Implications for Restarted PDHG
arxiv(2023)
摘要
There has been a recent surge in development of first-order methods (FOMs)
for solving huge-scale linear programming (LP) problems. The attractiveness of
FOMs for LP stems in part from the fact that they avoid costly matrix
factorization computation. However, the efficiency of FOMs is significantly
influenced - both in theory and in practice - by certain instance-specific LP
condition measures. Recently it was shown that the performance of the restarted
primal-dual hybrid gradient method (PDHG) is predominantly determined by two
specific condition measures: LP sharpness and Limiting Error Ratio. 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 Error Ratio involving
the reciprocal of the LP sharpness. This shows that in LP instances where there
is a dual nondegenerate optimal solution, the computational complexity of
restarted PDHG can be characterized solely in terms of LP sharpness and the
distance to optimal solutions, and simplifies the theoretical complexity upper
bound of restarted PDHG for these instances.
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