Coordinate-wise minimization is a simple popular method for large-scale optimization. Unfortunately, for general (non-differentiable and/or constrained) convex problems, its fixed points may not be global minima. We present two classes of linear programs (LPs) that coordinate-wise minimization solves exactly. We show that these classes subsume the dual LP relaxations of several well-known combinatorial optimization problems and the method finds a global minimum with sufficient accuracy in reasonable runtimes. Moreover, we experimentally show that the method frequently yields good suboptima or even optima for sparse LPs where optimality is not guaranteed in theory. Though the presented problems can be solved by more efficient methods, our results are theoretically non-trivial and can lead to new large-scale optimization algorithms in the future.

坐标最小化是一种用于大规模优化的简单流行方法。不幸的是，对于一般的（不可微和/或受约束的）凸问题，其固定点可能不是全局最小值。我们介绍了两类线性程序（LP），它们可以精确地解决坐标问题。我们表明，这些类包含了几个众所周知的组合优化问题的双重LP松弛，并且该方法在合理的运行时间中找到了具有足够精度的全局最小值。此外，我们通过实验证明，对于稀疏LP而言，该方法经常会产生良好的次优甚至最优，而理论上不能保证最优。尽管所提出的问题可以通过更有效的方法解决，但我们的结果在理论上并不重要，并且将来可能会导致新的大规模优化算法。