Generator Subadditive Functions for Mixed-Integer Programs

TL;DR

This paper investigates the theoretical properties of generator subadditive functions in the context of equality-constrained mixed-integer programs, establishing conditions for strong duality and exploring their applications.

Contribution

It extends the understanding of generator subadditive functions and strong duality to more general mixed-integer programs constrained by monoids and convex cones.

Findings

Strong duality holds under certain conditions for these MIPs.

Milder conditions for strong duality when monoids are defined by convex cones.

Examples demonstrate applications of the theoretical results.

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.