We consider the problem of finding the optimal diagonal preconditioner for a positive definite matrix. Although this problem has been shown to be solvable and various methods have been proposed, none of the existing approaches are scalable to matrices of large dimension, or when access is limited to black-box matrix-vector products, thereby significantly limiting their practical application. In view of these challenges, we propose practical algorithms applicable to finding approximate optimal diagonal preconditioners of large sparse systems. Our approach is based on the idea of dimension reduction, and combines techniques from semi-definite programming (SDP), random projection, semi-infinite programming (SIP), and column generation. Numerical experiments demonstrate that our method scales to sparse matrices of size greater than $10^7$. Notably, our approach is efficient and implementable using only black-box matrix-vector product operations, making it highly practical for a wide variety of applications.

翻译：我们研究正定矩阵的最优对角预条件子求解问题。尽管该问题已被证明可解且已有多种方法提出，但现有方法均无法扩展至高维矩阵，或在仅能通过黑盒矩阵-向量乘积访问矩阵时难以应用，这严重限制了其实际应用价值。针对这些挑战，我们提出了适用于大规模稀疏系统近似最优对角预条件子求解的实用算法。该方法基于降维思想，融合了半定规划（SDP）、随机投影、半无限规划（SIP）与列生成等技术。数值实验表明，我们的方法可扩展至维度超过$10^7$的稀疏矩阵。特别值得注意的是，该算法仅需黑盒矩阵-向量乘积操作即可高效实现，使其在众多实际应用场景中具备高度实用性。