The Homogenization Cone: Polar Cone and Projection

TL;DR

This paper investigates the properties of homogenization cones derived from convex sets in Hilbert spaces, focusing on their polar cones and an algorithm for projection, with applications to important cones like the second-order cone.

Contribution

It characterizes the polar cone of homogenization cones and proposes an algorithm for projection onto these cones using projections onto the original convex set.

Findings

Explicit description of the polar cone of the homogenization cone.

An algorithm for projecting onto the homogenization cone using projections onto the original set.

Illustrative examples demonstrating the theoretical results.

Abstract

Let $C$ be a closed convex subset of a real Hilbert space containing the origin, and assume that $K$ is the homogenization cone of $C$, i.e., the smallest closed convex cone containing $C \times \{1\}$. Homogenization cones play an important role in optimization as they include, for instance, the second-order/Lorentz/"ice cream" cone. In this note, we discuss the polar cone of $K$ as well as an algorithm for finding the projection onto $K$ provided that the projection onto $C$ is available. Various examples illustrate our results.