Submodular Function Minimization and Polarity

TL;DR

This paper introduces a polar-based outer approximation for the epigraph of set functions, demonstrating its exactness for submodular functions and its effectiveness in optimization tasks.

Contribution

It provides a novel polar relaxation approach that is equivalent to the Lovász extension and offers an alternative proof for the convex hull of submodular epigraphs.

Findings

Polar relaxation is exact for submodular functions

Outer approximations serve as effective cutting planes

Method improves optimization of submodular and non-submodular functions

Abstract

Using polarity, we give an outer polyhedral approximation for the epigraph of set functions. For a submodular function, we prove that the corresponding polar relaxation is exact; hence, it is equivalent to the Lov\'asz extension. The polar approach provides an alternative proof for the convex hull description of the epigraph of a submodular function. Computational experiments show that the inequalities from outer approximations can be effective as cutting planes for solving submodular as well as non-submodular set function minimization problems.