论文标题
沿原始射线偶联和支持功能的广义凸度恒定
Constant Along Primal Rays Conjugacies and Generalized Convexity for Functions of the Support
论文作者
论文摘要
r d中向量的支持是具有非零条目的索引集。支持的功能使该属性是0均匀的，因此，Fenchel偶联性无法提供相关的分析。在本文中，我们通过将经典的Fenchel标量产品耦合除以R d的给定（源）标准来定义R D和自身之间的耦合CAPRA。我们的主要结果是，当源规范及其双重标准都是矫形单调的时，支撑映射的任何无责任的有限值函数都是Capra-convex，即等于其Capra-biconjugate（广义凸）。我们还确定，任何此类功能都是在R d上具有适当凸的较低半连续功能的组成，其在单位球上的归一化映射（隐藏的凸度），并且在归一化时，它承认了一个变异配方，该配方涉及一个广义的本地K-Support双重辅助双重规范。
The support of a vector in R d is the set of indices with nonzero entries. Functions of the support have the property to be 0-homogeneous and, because of that, the Fenchel conjugacy fails to provide relevant analysis. In this paper, we define the coupling Capra between R d and itself by dividing the classic Fenchel scalar product coupling by a given (source) norm on R d. Our main result is that, when both the source norm and its dual norm are orthant-strictly monotonic, any nondecreasing finite-valued function of the support mapping is Capra-convex, that is, is equal to its Capra-biconjugate (generalized convexity). We also establish that any such function is the composition of a proper convex lower semi continuous function on R d with the normalization mapping on the unit sphere (hidden convexity), and that, when normalized, it admits a variational formulation, which involves a family of generalized local-K-support dual norms.
