通过度量投影范数的保序性和角距离的反序性刻画次对偶锥

中文摘要 AI 辅助

设$H$为实希尔伯特空间，$K\subseteq H$为闭凸锥，$K^*$为其对偶闭凸锥，$P_K$表示到$K$上的度量投影。我们研究与投影$P_K$相关的标量量及其与$K$诱导的预序的相互作用。主要结果通过投影范数的单调性刻画次对偶性：$K\subseteq K^*$当且仅当$x\le_K y$蕴含$\\|P_Kx\\|\le \\|P_Ky\\|$。我们还讨论投影范数及闭凸锥距离的次线性性，将这些函数与非对称半范数关联，并引入闭凸锥的归一化角距离。该角距离可通过$P_K$和普通距离函数得到等价公式，且对于满足$K\ne H$的闭凸锥，它给出次对偶性的第二种刻画。最后，利用有限维空间上的坐标wise单调范数聚合多个闭凸锥的距离，从而构造非对称半范数。所有证明均基于Moreau分解定理和度量投影的几何性质。

英文摘要

Let $H$ be a real Hilbert space, let $K\subseteq H$ be a closed convex cone, let $K^*$ be its dual closed convex cone, and let $P_K$ denote the metric projection onto $K$. We study scalar quantities associated with th projection $P_K$ and their interaction with the preorder induced by $K$. Our main result characterizes subduality by the monotonicity of the norm of a projection: $K\subseteq K^*$ if and only if $x\le_K y$ implies $\|P_Kx\|\le \|P_Ky\|$. We also discuss the sublinearity of the norm of a projection and of the distance from a closed convex cone, relate these functions to asymmetric seminorms, and introduce a normalized angular distance from a closed convex cone. The angular distance admits equivalent formulas in terms of $P_K$ and the ordinary distance function and, for closed convex cones with $K\ne H$, yields a second characterization of subduality. Finally, coordinatewise monotone norms on finite-dimensional spaces are used to aggregate distances from several closed convex cones and thereby construct asymmetric seminorms. The proofs are consequences of Moreau's decomposition theorem and the geometry of metric projections.

发表机构

• University of Birmingham（伯明翰大学）

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