Convex Analysis at Infinity: An Introduction to Astral Space. (arXiv:2205.03260v1 [math.OC])

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. …

analysis arxiv introduction math space