用于线性规划的重启 Halpern PDHG
Restarted Halpern PDHG for Linear Programming
AI总结：
提出无矩阵的 rHPDHG 及反射扩展 r²HPDHG 求解线性规划，证明其在可行、有界和不可行情形下的加速收敛与复杂度优势，GPU 实验在 383 个 MIPLIB 实例上优于 cuPDLP.jl。
摘要：
本文提出并分析了一种新的无矩阵原始-对偶算法，称为重启 Halpern 原始-对偶混合梯度（restarted Halpern primal-dual hybrid gradient，rHPDHG），用于求解线性规划（LP）。我们证明，rHPDHG 在可行且有界的 LP 上能够实现最优加速线性收敛。此外，我们给出了一项精细化分析，表明 rHPDHG 相对于普通 PDHG 具有加速的两阶段收敛性：其识别复杂度得到改进，并且最终的加速线性收敛不依赖于保守的全局 Hoffman 常数。关于不可行 LP，我们证明 rHPDHG 能够以加速线性速率恢复不可行性证书，改进了此前的收敛速率。进一步地，我们讨论了 rHPDHG 的一种扩展，通过加入反射操作（称为 r²HPDHG），并证明它保留 rHPDHG 的全部理论保证，同时在复杂度界中额外获得 2 倍加速。最后，我们使用 rHPDHG/r²HPDHG 构建了一个基于 GPU 的 LP 求解器；在 383 个 MIPLIB 实例上的实验表明，与 cuPDLP.jl 相比，其数值性能有所提升。
英文摘要：
In this paper, we propose and analyze a new matrix-free primal-dual algorithm, called restarted Halpern primal-dual hybrid gradient (rHPDHG), for solving linear programming (LP). We show that rHPDHG can achieve optimal accelerated linear convergence on feasible and bounded LP. Furthermore, we present a refined analysis that demonstrates an accelerated two-stage convergence of rHPDHG over the vanilla PDHG with an improved complexity for identification and an accelerated eventual linear convergence that does not depend on the conservative global Hoffman constant. Regarding infeasible LP, we show that rHPDHG can recover infeasibility certificates with an accelerated linear rate, improving the previous convergence rates. Furthermore, we discuss an extension of rHPDHG by adding reflection operation (which is dubbed as $\mathrm{r^2HPDHG}$), and demonstrate that it shares all theoretical guarantees of rHPDHG with an additional factor of 2 speedup in the complexity bound. Lastly, we build up a GPU-based LP solver using rHPDHG/$\mathrm{r^2HPDHG}$, and the experiments on 383 MIPLIB instances showcase an improved numerical performance compared cuPDLP.jl.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
