Distributional Lifting for Discrete Optimization

Abstract

We introduce a framework that transforms discrete optimization into continuous optimization over parameterized probability distributions. The framework reveals a common distributional structure behind several methods that have largely developed independently, including continuous reformulations for multilinear binary optimization, Markov decision processes and reinforcement learning, the cross-entropy method, mean-field variational optimization, and the quantum approximate optimization algorithm. It enables flexible and modular algorithm design, for example, by using neural networks to incorporate contextual information or by enforcing convexity of the lifted problem. We establish two theoretical results: a link between the lifted optimality gap and sampling quality in the original discrete problem, and a convergence guarantee for the lifted optimization problem under suitable conditions. These results lead to a unified algorithmic template based on batched sampling and stochastic gradient updates that is naturally compatible with GPU computation. We demonstrate the framework through three case studies spanning multidimensional knapsack, covering-constrained three-dimensional assignment, and a stochastic mixed-integer location-allocation problem, showing that it can accommodate distinct combinatorial structures and scale to large discrete and mixed-integer settings.