论文标题
无穷大的凸分析：星体空间简介
Convex Analysis at Infinity: An Introduction to Astral Space
论文作者
论文摘要
并非所有凸函数都在$ \ mathbb {r}^n $上具有有限的最小化器；有些只能通过序列最小化，直到无穷大。在这项工作中，我们旨在开发一种理论，以理解无穷大的这种最小化。我们研究星体空间，这是$ \ mathbb {r}^n $的紧凑型扩展，并添加了此类点。星体空间的构造是尽可能小的，同时仍确保所有线性函数都可以连续扩展到新空间。尽管星体空间包括所有$ \ mathbb {r}^n $，但它不是矢量空间，甚至不是公制空间。但是，它的结构足够良好，可以允许有用且有意义的分子性，结合性和细分差异的概念扩展。我们开发了这些概念，并分析了凸函数在星体空间上的各种特性，包括其最小化器的详细结构，连续性的确切特征和下降算法的收敛性。
Not all convex functions on $\mathbb{R}^n$ have finite minimizers; some can only be minimized by a sequence as it heads to infinity. In this work, we aim to develop a theory for understanding such minimizers at infinity. We study astral space, a compact extension of $\mathbb{R}^n$ to which such points at infinity have been added. Astral space is constructed to be as small as possible while still ensuring that all linear functions can be continuously extended to the new space. Although astral space includes all of $\mathbb{R}^n$, it is not a vector space, nor even a metric space. However, it is sufficiently well-structured to allow useful and meaningful extensions of concepts of convexity, conjugacy, and subdifferentials. We develop these concepts and analyze various properties of convex functions on astral space, including the detailed structure of their minimizers, exact characterizations of continuity, and convergence of descent algorithms.
