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

Debashis Chatterjee

TL;DR

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 is developed.

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.

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

TL;DR

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 is developed.

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.
Paper Structure (71 sections, 7 theorems, 69 equations, 23 figures, 6 tables, 1 algorithm)

This paper contains 71 sections, 7 theorems, 69 equations, 23 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Research objective.
  3. Novelty and contribution.
  4. Related Work
  5. Chance-constrained optimization.
  6. Robust optimization.
  7. Scenario-based optimization.
  8. Convex approximations of chance constraints.
  9. Bayesian decision theory and optimization.
  10. Contribution relative to existing literature.
  11. Problem setup
  12. Bayesian learning of coefficients
  13. Decision criterion
  14. Posterior feasibility
  15. Methodology
  16. ...and 56 more sections

Key Result

Proposition 4.1

If $x$ satisfies eq:robustCred and $\mathcal{C}_{1-\alpha}(D)$ satisfies eq:credmass, then $x$ is $(1-\alpha)$ posterior-feasible, i.e. $V_D(x)\le \alpha$.

Figures (23)

  • Figure 3: Risk calibration: achieved true violation versus target $\alpha$ (ideal calibration corresponds to the diagonal).
  • Figure 4: Profit--risk trade-off across all trials pooled: scatter of profit versus true violation probability.
  • Figure 5: Posterior safety certificate: mean conservative upper bound $\widehat{V}_{\text{post,UB95}}$ versus target $\alpha$.
  • Figure 6: PBMC3k UMAP embedding colored by the 8 provided cluster labels.
  • Figure 7: Discrimination weights $w_g$ for the selected panel genes (larger is more between-cluster separation).
  • ...and 18 more figures

Theorems & Definitions (20)

  • Definition 3.1: Posterior feasibility and violation
  • Remark 3.1: Interpretation
  • Proposition 4.1: Credible-set robustification implies posterior feasibility
  • proof
  • Theorem 4.1: Posterior-scenario violation bound (scenario approach)
  • Remark 4.1
  • Corollary 4.1: Choosing $N$ for a target posterior-feasibility level
  • proof : Proof of Proposition \ref{['prop:credImplies']}
  • Proposition A.1: SOC reformulation of an ellipsoidal robust linear constraint
  • proof
  • ...and 10 more