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Generator Subadditive Functions for Mixed-Integer Programs

Gustavo Ivan Angulo Olivares, Burak Kocuk, Diego Moran Ramirez

Abstract

For equality-constrained linear mixed-integer programs (MIP) defined by rational data, it is known that the subadditive dual is a strong dual and that there exists an optimal solution of a particular form, termed generator subadditive function. Motivated by these results, we explore the connection between Lagrangian duality, subadditive duality and generator subadditive functions for general equality-constrained MIPs where the vector of variables is constrained to be in a monoid. We show that strong duality holds via generator subadditive functions under certain conditions. For the case when the monoid is defined by the set of all mixed-integer points contained in a convex cone, we show that strong duality holds under milder conditions and over a more restrictive set of dual functions. Finally, we provide some examples of applications of our results.

Generator Subadditive Functions for Mixed-Integer Programs

Abstract

For equality-constrained linear mixed-integer programs (MIP) defined by rational data, it is known that the subadditive dual is a strong dual and that there exists an optimal solution of a particular form, termed generator subadditive function. Motivated by these results, we explore the connection between Lagrangian duality, subadditive duality and generator subadditive functions for general equality-constrained MIPs where the vector of variables is constrained to be in a monoid. We show that strong duality holds via generator subadditive functions under certain conditions. For the case when the monoid is defined by the set of all mixed-integer points contained in a convex cone, we show that strong duality holds under milder conditions and over a more restrictive set of dual functions. Finally, we provide some examples of applications of our results.

Paper Structure

This paper contains 15 sections, 16 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Duality for equality-constrained mixed-integer programs
  3. Lagrangian duality and generator subadditive functions
  4. Subadditive duality
  5. Duality for equality-constrained conic mixed-integer programs
  6. Strong duality
  7. Conic MIPs where integer and continuous variables come from different cones
  8. Conic MIPs with block structure
  9. Conic integer programs defined by cones with $(R,G)$-finitely generated integral vectors
  10. Illustrations of our results
  11. A numerical example
  12. Examples of sets ${\mathcal{M}}^\leq(b)$ with conic representable integer hull
  13. Conic sets with finitely many integer fibers
  14. Packing conic mixed-integer programs
  15. An example of a conic MIP with block structure: clustering problem

Key Result

Proposition 1

Consider a feasible linear MIP (i.e., ${\mathcal{M}}= \mathbb{Z}_+^{n_1} \times \mathbb{R}_+^{n_2}$) of the form eq:genX_MIP, where $A\in {\mathbb{Q}}^{m \times n_1}$, $G\in {\mathbb{Q}}^{m \times n_2}$ and $b \in {\mathbb{Q}}^m$, and the subadditive dual as defined in eq:subadditiveDualLinear. The

Figures (1)

  • Figure 1: $F_{10445}$ vs. $F'_{10445}$ over $\Omega$.

Theorems & Definitions (34)

  • Proposition 1: johnson1974groupjeroslow1978cutting
  • Proposition 2: Klabjan2007cheung2016certificates
  • Proposition 3: Klabjan2007cheung2016certificates
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Definition 1: Generator subadditive function
  • Definition 2: Subadditive function
  • Definition 3: Integral generating set
  • ...and 24 more