Blockwise coordinate descent methods have a long tradition in continuous optimization and are also frequently used in discrete optimization under various names. New interest in blockwise coordinate descent methods arises for improving sequential solutions for problems which consist of several planning stages. In this paper we systematically formulate and analyze the blockwise coordinate descent method for integer programming problems. We discuss convergence of the method and properties of the resulting solutions. We extend the notion of Pareto optimality for blockwise coordinate descent to the case that the blocks do not form a partition and compare Pareto optimal solutions to blockwise optimal and to global optimal solutions. Among others we derive a condition which ensures that the solution obtained by blockwise coordinate descent is weakly Pareto optimal and we confirm convergence of the blockwise coordinate descent to a global optimum in matroid polytopes. The results are interpreted in the context of multi-stage linear integer programming problems and illustrated for integrated planning in public transportation.

中文翻译：

---

整数程序的分块坐标下降法

逐块坐标下降法在连续优化中有着悠久的历史，并且在各种名称下也经常用于离散优化中。对于改进由多个计划阶段组成的问题的顺序解决方案，人们对块状坐标下降法产生了新的兴趣。在本文中，我们系统地制定和分析了整数规划问题的逐块坐标下降法。我们讨论了方法的收敛性以及所得解决方案的性质。我们将用于块状坐标下降的帕累托最优概念扩展到块不形成分区的情况，并将帕累托最优解与块状最优解和全局最优解进行比较。除其他外，我们推导了一个条件，该条件确保通过块状坐标下降获得的解是弱的Pareto最优解，并且我们确定在类拟阵多面体中，块状坐标下降到全局最优解的收敛性。在多阶段线性整数规划问题的背景下解释了结果，并为公共交通的综合规划说明了结果。