Generalized Affine Programming & Duality Gap with non-Division Rings

TL;DR

This paper extends classical affine programming to ordered rings beyond real vector spaces, analyzing duality properties and identifying conditions under which duality theorems fail, especially over rings like integers.

Contribution

It generalizes affine programming to ordered rings, classifies rings where duality theorems fail, and explores the impact on primal-dual relationships.

Findings

Classical duality theorems hold over real vector spaces.

Strong duality fails over the integers.

Certain ordered rings preserve duality properties.

Abstract

Classical primal-dual affine programming takes place over finite dimensional real vector spaces. This results in beautiful duality theory, connecting the optimal solu- tions of the primal maximization problem and the dual minimization problems. These results include the Existence Duality Theorem, which guarantees optimal solutions to any feasible bounded program; and the Strong Duality Theorem, which implies that optimal solutions for primal and dual programs must have the same objective value. In a common extension of classical affine programming, we see that the Strong Duality does not hold when ring of scalars is the integers. Extension of classical affine programming results to ordered division rings are explored in. In this paper, we describe the generalized setting of affine programming using ordered ring (not necessarily division), and classify the rings for which the Existence Duality Theorem or the Strong Duality Theorem fail.