A Characterization of Simultaneous Optimization, Majorization, and (Bi-)Submodular Polyhedra
受电源管理资源分配问题启发，研究了同时最小化Schur凸函数（即最小主化元素）的解的存在性，引入(a,b)-主化概念，并用基多面体和（双）子模多面体刻画了具有此类元素的可行集，扩展了经典贪婪算法刻画，并讨论了在电源管理、凸合作博弈和正则化回归中的应用。
Motivated by resource allocation problems (RAPs) in power management applications, we investigate the existence of solutions to optimization problems that simultaneously minimize the class of Schur-convex functions, also called least-majorized elements. For this, we introduce a generalization of majorization and least-majorized elements, called (a, b)-majorization and least (a, b)-majorized elements, and characterize the feasible sets of problems that have such elements in terms of base and (bi-)submodular polyhedra. Hereby, we also obtain new characterizations of these polyhedra that extend classical characterizations in terms of optimal greedy algorithms from the 1970s. We discuss the implications of our results for RAPs in power management applications and derive a new characterization of convex cooperative games and new properties of optimal estimators of specific regularized regression problems. In general, our results highlight the combinatorial nature of simultaneously optimizing solutions and provide a theoretical explanation for why such solutions generally do not exist.