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On Integer Programs with Irrational Data

Seyedmohammadhossein Hosseinian, Andrew J. Schaefer

Abstract

An integer program (IP) with a finite number of feasible solutions may have an unbounded linear programming relaxation if it contains irrational parameters, due to implicit constraints enforced by the irrational numbers. We show that those constraints can be obtained if the irrational parameters are polynomials of roots of integers over the field of rational numbers, leading to an equivalent rational formulation. We also establish a weaker result for IPs involving the general class of algebraic irrational parameters, which extends to IPs with a particular form of transcendental numbers.

On Integer Programs with Irrational Data

Abstract

An integer program (IP) with a finite number of feasible solutions may have an unbounded linear programming relaxation if it contains irrational parameters, due to implicit constraints enforced by the irrational numbers. We show that those constraints can be obtained if the irrational parameters are polynomials of roots of integers over the field of rational numbers, leading to an equivalent rational formulation. We also establish a weaker result for IPs involving the general class of algebraic irrational parameters, which extends to IPs with a particular form of transcendental numbers.
Paper Structure (3 sections, 4 theorems, 15 equations)

This paper contains 3 sections, 4 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. IPs with Irrational Parameters

Key Result

Theorem 2

byrd1987recognizing Let $\mathcal{P} = \{{\bf x} \in \mathbb{R}^n : {\bf A x = b}, \, {\bf x \geq 0}\}$, and $\mathcal{S}$ be the convex hull of the integer solutions in $\mathcal{P}$. If $\mathcal{P}$ is unbounded, yet contains only a finite (but nonempty) set of integer solutions, then it is of a

Theorems & Definitions (6)

  • Example 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Corollary 5