Circumcentric directions of cones

TL;DR

This paper introduces circumcentric directions for convex cones, showing they point inward and can generate feasible search directions, potentially improving projection methods for convex feasibility problems.

Contribution

It extends the concept of circumcenters to convex cones, proving they lie in the interior of polars and can be used to find feasible directions.

Findings

Circumcentric directions belong to the interior of cone polars.

They provide feasible search directions within convex regions.

A measure of interiorness for these directions is derived.

Abstract

Generalized circumcenters have been recently introduced and employed to speed up classical projection-type methods for solving feasibility problems. In this note, circumcenters are enforced in a new setting; they are proven to provide inward directions to sets given by convex inequalities. In particular, we show that circumcentric directions of finitely generated cones belong to the interior of their polars. We also derive a measure of interiorness of the circumcentric direction, which then provides a special cone of search directions, all being feasible to the convex region under consideration.