发表机构
National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种多团分解方法McADMM，用于高效求解大规模稀疏SDP，通过减少投影成本并支持GPU加速，在QCQP松弛问题上优于现有求解器。
AI 中文摘要
sGS-PADMM [21, 14, 6] 是一类功能强大且用途广泛的可收敛多块 ADMM 求解器，适用于中等规模的线性半定规划（SDP）问题。在本文中，我们通过提出一种新的多团分解方法，进一步增强此类算法以求解 SDP 问题，使得在具有有利的聚合稀疏模式的大规模稀疏 SDP（例如，n > 1000）应用中能够获得显著改进。我们的 SDP 分解策略主要旨在减少分解后的估计 PSD 投影成本，这与常见的以最小化团间重叠为目标的分解算法不同。这一特性得益于我们新颖的线性空间投影方法，该方法能够通过简单的平均步骤高效处理大量重叠约束。在数值实验中，我们展示了我们的求解器——名为 McADMM（多团 ADMM）——在由一些重要的二次约束二次规划（QCQP）问题的松弛产生的大量大规模 SDP 实例上的性能。我们将 McADMM 的性能与其他最先进的基于分解的求解器以及未分解的 sGS-PADMM 进行对比，以突出我们的关键贡献。此外，我们还开发了 McADMM 的 GPU 实现，并证明它可以显著加速分解求解器。
英文摘要
sGS-PADMM [21, 14, 6] is a powerful and versatile class of convergent multi-block ADMM solvers for implementations on moderate-sized linear semidefinite programming (SDP) problems. In this paper, we further enhance this class of algorithms for solving SDP problems by proposing a new multi-clique decomposition approach, allowing substantial improvements in applications on large-scale sparse SDPs (e.g., where $n > 1000$) with conducive aggregate sparsity patterns. Our SDP decomposition strategy mainly aims to reduce the estimated PSD projection cost after decomposition, in contrast to common decomposition algorithms that are encumbered with minimizing the overlaps between cliques. This feature is made possible by our novel linear-space projection approach that is capable of efficiently processing a large number of overlap constraints via simple averaging steps. For the numerical experiments, we demonstrate the performance of our solver -- named McADMM for Multi-clique ADMM -- on a number of large-scale SDP instances that arise from relaxations of some important quadratically constrained quadratic programming (QCQP) problems. The performance of McADMM is contrasted against other state-of-the-art decomposition-based solvers as well as the non-decomposed sGS-PADMM to highlight our key contributions. We additionally develop a GPU implementation of McADMM and demonstrate that it can substantially accelerate the decomposed solver.
Comments38 pages, 2 figures
