We consider applications involving a large set of instances of projecting points to polytopes. We develop an intuition guided by theoretical and empirical analysis to show that when these instances follow certain structures, a large majority of the projections lie on vertices of the polytopes. To do these projections efficiently we derive a vertex-oriented incremental algorithm to project a point onto any arbitrary polytope, as well as give specific algorithms to cater to simplex projection and polytopes where the unit box is cut by planes. Such settings are especially useful in web-scale applications such as optimal matching or allocation problems. Several such problems in internet marketplaces (e-commerce, ride-sharing, food delivery, professional services, advertising, etc.), can be formulated as Linear Programs (LP) with such polytope constraints that require a projection step in the overall optimization process. We show that in the very recent work, the polytopic projection is the most expensive step and our efficient projection algorithms help in gaining massive improvements in performance.
翻译:我们考虑的是涉及大量多面体投影点的应用程序。我们根据理论和经验分析开发了一种直觉,以表明当这些实例遵循某些结构时,大多数预测都发生在多面体的顶端。为了高效地进行这些预测,我们得出了一个面向顶点的递增算法,将一个点投到任何任意的多面体上,并给出特定的算法,以迎合简单X投影和多面方块被飞机切割的组合框。这些设置在网络规模的应用中特别有用,例如最佳匹配或分配问题。互联网市场中的一些问题(电子商务、搭载、食品供应、专业服务、广告等)可以作为线形方案(LP),而这种多面方案要求在整个优化过程中采取预测步骤。我们表明,在最近的工作中,多面投影法是最昂贵的一步,我们高效的预测算法有助于大规模地改进性能。