Characterization and Approximation of Strong General Dual Feasible Functions

TL;DR

This paper explores the properties of dual feasible functions (DFFs), characterizes their maximal forms, and demonstrates how they can be approximated by extreme DFFs, advancing their application in optimization bounds.

Contribution

It provides a new characterization of maximal general DFFs, proves a 2-slope theorem for extreme DFFs, and shows approximation methods for restricted maximal DFFs.

Findings

Characterization of (restricted/strongly) maximal general DFFs.

Proof of a 2-slope theorem for extreme general DFFs.

Any restricted maximal general DFF can be approximated by an extreme DFF.

Abstract

Dual feasible functions (DFFs) have been used to provide bounds for standard packing problems and valid inequalities for integer optimization problems. In this paper, the connection between general DFFs and a particular family of cut-generating functions is explored. We find the characterization of (restricted/strongly) maximal general DFFs and prove a 2-slope theorem for extreme general DFFs. We show that any restricted maximal general DFF can be well approximated by an extreme general DFF.