Three different views on barrier functions in conic optimization
Keywords: conic optimization, self-concordance, self-scaledness, canonical barrier, affine differential geometry, cubic form, minimal surface, Lagrangian immersion, parallel derivative
TL;DR: Besides the familiar analytic view-point on barriers in conic optimization, these objects can be equivalently considered from two different geometric view-points, affine differential geometry and the geometry of Larangian immersions.
Abstract: A smooth enough function $f: D \to \mathbb R$ defined on a domain $D \subset V$ in a real vector space is an analytic object which defines two different geometric objects. On the one hand, one may consider its level hypersurfaces as affine hypersurface immersions in $V$, on the other hand the graph of its gradient $\nabla f$ as a Lagrangian immersion in the product $V \times V^*$ of the real space with its dual. It turns out that when $f$ is a logarithmically homogeneous barrier on a conic domain $D$, then the three objects are almost equivalent to each other in the sense that they contain nearly the same information and can be recovered from each other. Properties of the barrier such as self-concordance and self-scaledness are equivalent to meaningful properties of the geometric objects. This equivalence furnishes new vantage points to study barriers in conic optimization and builds a bridge to other areas of mathematics which open new ways to obtain results in optimization. We describe the links and equivalences between the three different view-points and give examples of results obtained by means of these connections.
Submission Number: 70
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