AI 中文总结
本文提出融入对数障碍函数的BPDHG算法，还开发BPDLP方法，实验显示其可缓解KKT残差平台期，且存在经验指标可识别BPDLP更具优势的LP问题。
AI 中文摘要
原始对偶混合梯度（PDHG）方法已被证实呈现两阶段收敛行为，其中持续的活动集识别阶段可能是收敛缓慢的主要问题。本文提出障碍PDHG（BPDHG），这是一种嵌套算法，将对数障碍函数融入PDHG框架以缓解该问题。我们首先建立内迭代的收敛性，推导对应内问题的误差界；随后证明BPDHG生成的外序列趋近于所考虑线性规划（LP）问题的最优解集。此外，我们将障碍技术集成到原始对偶线性规划（PDLP）框架中，开发对应障碍PDLP（BPDLP）方法。数值实验表明，障碍修改可缓解选定实例中KKT残差的持续平台期；我们还研究了一种经验性的实例相关指标，用于识别BPDLP更可能优于PDLP的LP问题。
英文摘要
Primal Dual Hybrid Gradient (PDHG) method has been verified to exhibit a two stage convergence behavior, in which a prolonged active set identification phase may be a major issue of slow convergence. In this paper, we propose Barrier PDHG (BPDHG), a nested algorithm which incorporates a logarithmic barrier function into the PDHG framework to alleviate this problem. We first establish convergence of the inner iterations, derive an error bound for the corresponding inner problem. Then we prove that the outer sequence generated by BPDHG approaches the optimal solution set of the LP problem we considered. Furthermore, we integrate the barrier technique into the {Primal Dual Linear Programming} (PDLP) framework to develop the corresponding Barrier PDLP (BPDLP) method. Numerical experiments show that the barrier modification can alleviate prolonged plateaus in the KKT residual on selected instances. We also investigate an empirical instance-dependent indicator for identifying LP problems on which BPDLP is more likely to outperform PDLP.
Comments41 pages, 12 figures
