Optimal Diagonal Preconditioning
将最优对角预条件问题建模为拟凸优化和半定规划，提出高效算法求解大规模稀疏矩阵的最优对角预条件，发现常用启发式方法与最优解存在显著差距。
A New Practical Algorithm Enables Optimal Preconditioning A classic and important question in optimization and numerical methods is how to find diagonal preconditioners to maximally reduce the condition number of any matrix with full rank. Until recently, few practical methods could handle large sparse matrices. In “Optimal Diagonal Preconditioning,” the authors show that this problem can be modeled using quasiconvex optimization and semidefinite programming. Leveraging these insights, they develop algorithms to efficiently find optimal diagonal preconditioners for large sparse systems. They find that although heuristic diagonal preconditioners are popular in practice, their performance at reducing condition numbers could have a significant gap to optimal diagonal preconditioners. This work provides theoretical foundation for future works on optimal preconditioning as well as practical implementations that could be used to build more sophisticated software for optimal preconditioning at scale. A main advantage of the framework in “Optimal Diagonal Preconditioning” is its potential to be scaled up to handle even larger matrices, which is an exciting direction for numerical optimization.