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Generalized Affine Programming & Duality Gap with non-Division Rings

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.