AI 中文总结
本文分析重启原对偶混合梯度法（rPDHG）在锥线性优化中的理论性能，建立原对偶水平集几何与收敛速度的关联，并指出其性能受水平集直径与锥半径之比影响。
AI 中文摘要
我们考虑在矩阵分解免方法具有吸引力或必要性的规模下求解（凸）锥线性优化问题。重启的原对偶混合梯度方法（rPDHG）——结合启发式增强和GPU实现——在求解超大规模线性优化问题（LP）方面已非常成功。然而，其在更一般的锥凸优化问题中的应用研究尚不充分。我们分析了rPDHG在一般（凸）锥线性优化及其特例LP中的理论性能。我们展示了原对偶δ-（子）水平集W_δ的几何性质与rPDHG收敛速度之间的关系。具体而言，我们证明了rPDHG收敛速度的一个界，当存在一个原对偶（子）水平集W_δ满足以下条件时，该界得到改进：（i）W_δ在Hausdorff距离下接近最优解集，（ii）W_δ的直径与“锥半径”之比很小。在LP的特殊情况下，rPDHG的性能仅受该比值应用于对应于最佳非最优极值点的（子）水平集的限制。根据问题实例的不同，该比值可能取极端值，导致rPDHG在理论和实践中的性能可能极好或极差。
英文摘要
We consider solving (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) -- with heuristic enhancements and GPU implementation -- has been very successful in solving huge-scale linear optimization problems (LPs). However, its application to more general conic convex optimization problems is not so well-studied. We analyze the theoretical performance of rPDHG for general (convex) conic linear optimization, and LP as a special case thereof. We show a relationship between the geometry of the primal-dual $δ$-(sub-)level sets ${W}_δ$ and the convergence rate of rPDHG. Specifically, we prove a bound on the convergence rate of rPDHG that improves when there is a primal-dual (sub-)level set ${W}_δ$ for which (i) ${W}_δ$ is close to the optimal solution set in Hausdorff distance, and (ii) the ratio of the diameter to the ``conic radius'' of ${W}_δ$ is small. And in the special case of LP, the performance of rPDHG is bounded only by this ratio applied to the (sub-)level set corresponding to the best non-optimal extreme point. Depending on the problem instance, this ratio can take on extreme values and can result in excellent or poor performance of rPDHG both in theory and in practice.
Comments35 pages, 6 figures
