Mean field variational framework for integer optimization

TL;DR

This paper introduces a mean field variational approach for solving constrained integer optimization problems, transforming them into continuous problems and demonstrating effectiveness especially at large scales.

Contribution

It presents a novel mean field variational framework that generalizes to diverse problem structures and scales better than traditional methods for large instances.

Findings

Comparable to specialized methods on small to medium problems

Capable of finding feasible solutions for large instances within reasonable times

Remains valid across diverse problem structures

Abstract

A principled method to obtain approximate solutions of general constrained integer optimization problems is introduced. The approach is based on the calculation of a mean field probability distribution for the decision variables which is consistent with the objective function and the constraints. The original discrete task is in this way transformed into a continuous variational problem. In the context of particular problem classes at small and medium sizes, the mean field results are comparable to those of standard specialized methods, while at large sized instances is capable to find feasible solutions in computation times for which standard approaches can't find any valuable result. The mean field variational framework remains valid for widely diverse problem structures so it represents a promising paradigm for large dimensional nonlinear combinatorial optimization tasks.