[Paper Review] Another Proof of the Fact that Polyhedral Cones are Finitely Generated

Volker Kaibel|ArXiv.org|Dec 15, 2009

graph theory and CDMA systems1 references3 citations

TL;DR

This paper presents a new inductive proof that every polyhedral cone is finitely generated, using matrix determinants and kernel dimension analysis. It shows that generators can be explicitly constructed from subdeterminants of the defining inequality matrix, ensuring rationality and polynomial encoding length.

ABSTRACT

In this note, we work out a simple inductive proof showing that every polyhedral cone K is the conic hull of a finite set X of vectors. The base cases of the induction are linear subspaces and linear halfspaces of linear subspaces. The proof also shows that the components of the vectors in X can be chosen (up to their sign) to be quotients of subdeterminants of the coefficient matrix of any inequality system defining K.

Motivation & Objective

• To provide a simple, inductive proof that every polyhedral cone is finitely generated, avoiding reliance on geometric notions like faces or dimension.
• To establish that the generators of a polyhedral cone can be explicitly constructed from quotients of subdeterminants of the coefficient matrix.
• To show that these generators lie in Δ(A)^n, where Δ(A) consists of quotients of subdeterminants of A, ensuring rationality and polynomial encoding length.
• To demonstrate that the proof extends to the full Weyl-Minkowski theorem by handling both bounded and unbounded cases via induction on the number of inequalities.
• To provide a constructive, elementary proof that supports the polynomial solvability of linear programming and the NP-membership of integer programming feasibility.

Proposed method

• Use induction on the number of inequalities in the system defining the polyhedral cone, starting from linear subspaces and halfspaces.
• Apply Lemma 1 to handle two cases: when the kernel intersection has high dimension (leading to finite generation via basis vectors), and when a vector z exists with z, -z ∉ K.
• For the latter case, use the orthogonal complement of the row sum space of the inequality matrix to find such a z, leveraging orthogonality and sign conditions.
• Construct generators by intersecting the cone with lower-dimensional subspaces defined by setting one inequality to equality, reducing the problem inductively.
• Show that any point x in the cone can be expressed as a conic combination of points from the union of generating sets of these lower-dimensional cones.
• Use the existence of λ* ≥ 0 such that x + λ*z ∈ K_i^*, and similarly for -z, to express x as a conic combination of two points in the union of generating sets.

Experimental results

Research questions

• RQ1 Can a simple, inductive proof be constructed to show that every polyhedral cone is finitely generated without relying on geometric concepts like faces?
• RQ2 What is the explicit structure of the finite generating set for a polyhedral cone defined by a system Ax ≤ 0?
• RQ3 Can the components of the generators be bounded in terms of subdeterminants of the coefficient matrix A?
• RQ4 How does the kernel dimension of the constraint matrix influence the existence of a finite generating set?
• RQ5 What conditions ensure that a vector z exists such that z and -z are not in the cone, enabling the inductive construction?

Key findings

• Every polyhedral cone defined by Ax ≤ 0 is the conic hull of a finite set X ⊆ Δ(A)^n, where Δ(A) consists of quotients of subdeterminants of A.
• The proof establishes that the generators can be chosen such that their components are quotients of subdeterminants of A, up to sign.
• The inductive proof avoids geometric notions by basing induction on the number of inequalities, with base cases being linear subspaces and linear halfspaces.
• When the kernel intersection has dimension at least dim(ker(C)) - 1, the cone is generated by a finite set derived from a basis of the kernel.
• If no such high-dimensional kernel intersection exists, a vector z exists such that z, -z ∉ K, enabling the construction of generators via conic combinations.
• The construction ensures that any point in the cone can be written as a conic combination of points from the union of generating sets of lower-dimensional cones, proving finite generation.