# Majorisation-minimisation algorithms for minimising the difference between lattice submodular functions

## TL;DR

This paper introduces majorisation-minimisation algorithms for optimizing the difference between lattice submodular functions, ensuring monotonic convergence to local minima and broad applicability.

## Contribution

It develops novel algorithms analogous to existing routines for lattice submodular functions and extends hardness results to a wider class of functions.

## Key findings

- Algorithms guarantee monotonic decrease in objective
- Convergence to local minima is proven
- Many functions can be expressed as differences of submodular functions

## Abstract

We consider the problem of minimising functions represented as a difference of lattice submodular functions. We propose analogues to the SupSub, SubSup and ModMod routines for lattice submodular functions. We show that our majorisation-minimisation algorithms produce iterates that monotonically decrease, and that we converge to a local minimum. We also extend additive hardness results, and show that a broad range of functions can be expressed as the difference of submodular functions.