Astral Space: Convex Analysis at Infinity
This work addresses a foundational issue in convex analysis for mathematicians and theorists, offering a new framework for handling infinite minimizers, though it appears incremental in extending existing convex analysis tools.
The paper tackles the problem of convex functions without finite minimizers by developing a theory for minimizers at infinity, introducing astral space as a compact extension of ℝⁿ to analyze such points. It extends concepts like convexity and subdifferentials to this space, providing characterizations of continuity and algorithm convergence.
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.