$MC^2$ Mixed Integer and Linear Programming

2025-11-25

AI Summary

This work addresses mixed-integer linear programming (MILP) and stochastic optimization problems by proposing a probabilistic solution framework grounded in the Boltzmann distribution. The original problem is reformulated as a Monte Carlo optimization task—sampling from truncated multivariate exponential and Gaussian distributions over the feasible constraint set—and solved efficiently via the Kent–Davis sampling algorithm. Unlike conventional deterministic solvers, this approach avoids strong structural assumptions on the problem, thereby enhancing scalability and stochastic exploration capability. Experiments on portfolio optimization and the canonical stochastic farmer problem demonstrate that the method achieves solution quality comparable to state-of-the-art commercial solvers (e.g., Gurobi) on medium-scale instances, while exhibiting superior robustness to high-dimensional, non-convex, or black-box constraints. The framework introduces a novel probabilistic modeling and optimization paradigm for MILP, bridging statistical sampling theory with discrete and stochastic decision-making.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Reformulating integer programming as Monte Carlo optimization problems
• Solving optimization through simulation of truncated multivariate distributions
• Applying methodology to portfolio optimization and stochastic programming problems

Innovation

• Reformulates optimization as Monte Carlo simulation
• Simulates from truncated exponential and Gaussian distributions
• Applies Kent and Davis algorithms for efficient sampling