An algorithm for minimum cardinality generators of cones

TL;DR

This paper introduces a polynomial-time algorithm for finding minimum cardinality generators of convex cones, extending known results from linear spaces to general cones through a novel decomposition approach.

Contribution

It provides the first polynomial-time method for computing minimum cardinality generators of finitely generated cones, improving upon previous algorithms that only find conically independent generators.

Findings

Proves the size of conically independent generators is at most twice the minimum cardinality.

Extends known linear space results to general convex cones via decomposition.

Develops a constructive, polynomial-time algorithm for minimum generator computation.

Abstract

This paper presents a novel proof that for any convex cone, the size of conically independent generators is at most twice that of minimum cardinality generators. While this result is known for linear spaces, we extend it to general cones through a decomposition into linear and pointed components. Our constructive approach leads to a polynomial-time algorithm for computing minimum cardinality generators of finitely generated cones, improving upon existing methods that only compute conically independent generators.