$MC^2$ Mixed Integer and Linear Programming

TL;DR

This paper introduces $MC^2$ algorithms that reformulate mixed integer and linear programming as Monte Carlo optimization problems, enabling efficient simulation-based solutions for complex constrained optimization tasks.

Contribution

The paper presents a novel Monte Carlo-based framework for solving mixed integer and linear programming problems using truncated distribution simulation techniques.

Findings

Effective simulation from truncated distributions achieved

Demonstrated on portfolio optimization and farmer problem

Provides new directions for future research in optimization

Abstract

In this paper, we design $MC^2$ algorithms for Mixed Integer and Linear Programming. By expressing a constrained optimisation as one of simulation from a Boltzmann distribution, we reformulate integer and linear programming as Monte Carlo optimisation problems. The key insight is that solving these optimisation problems requires the ability to simulate from truncated distributions, namely multivariate exponentials and Gaussians. Efficient simulation can be achieved using the algorithms of Kent and Davis. We demonstrate our methodology on portfolio optimisation and the classical farmer problem from stochastic programming. Finally, we conclude with directions for future research.