Bayesian Linear Programming under Learned Uncertainty: Posterior Feasibility Guarantees, Scenario Certification, and Applications

2026-03-06

AI Summary

This work addresses linear programming problems where both constraint and objective coefficients must be learned from data, proposing the first Bayesian linear programming framework that models parameter uncertainty via probability distributions and updates posteriors using observed data. The approach integrates two strategies—trust-region robust optimization and posterior scenario sampling—to embed uncertainty-aware decisions, and introduces Monte Carlo certification to quantify residual infeasibility. Key contributions include a posterior feasibility guarantee mechanism, an interpretable scenario optimization method with finite-sample guarantees, and a data-driven diagnostic procedure for infeasibility. Experiments demonstrate that the proposed method significantly outperforms plug-in estimators in safety-critical performance and yields scientifically interpretable, uncertainty-aware decisions on single-cell transcriptomic data.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Linear programming is widely used for decision-making in science, engineering, and operations research, yet in many modern applications the coefficients entering the constraints and objective are not known exactly and must be learned from data. Classical stochastic and robust optimization offer two influential paradigms for handling such uncertainty, but they typically treat the underlying uncertainty description as given and do not directly integrate priors and updated to posteriors guarantees. This paper develops a Bayesian framework for linear programming in which uncertain quantities are modeled probabilistically, updated through observed data, and propagated into optimization through posterior feasibility requirements. We present two complementary computational strategies: a credible-region robustification that converts posterior uncertainty into deterministic protection, and a posterior-scenario approach that uses sampled posterior realizations to construct tractable optimization problems with finite-sample interpretability. We also propose a Monte Carlo certification procedure that provides conservative, data-conditioned assessments of residual infeasibility. Simulation experiments show that the proposed framework substantially improves safety relative to naive plug-in decisions, while a real-data study on single-cell transcriptomic data demonstrates that the approach can produce scientifically interpretable decisions together with explicit uncertainty-aware feasibility diagnostics. The proposed methodology offers a unified bridge between Bayesian learning, optimization under uncertainty, and practical decision certification.

Problem

• Bayesian optimization
• linear programming
• uncertainty quantification
• posterior feasibility
• data-driven decision-making

Innovation

• Bayesian optimization
• posterior feasibility
• linear programming under uncertainty
• scenario approximation
• Monte Carlo certification