Optimal Policy Learning under Budget and Coverage Constraints

TL;DR

This paper develops a framework for optimal policy learning considering budget and coverage constraints, demonstrating a knapsack structure and proposing algorithms with near-optimal performance supported by Monte Carlo experiments.

Contribution

It introduces a novel characterization of the optimal policy under combined constraints and analyzes two implementable algorithms with theoretical and empirical guarantees.

Findings

The optimal policy can be characterized by an affine threshold rule.

The linear programming relaxation has an O(1) integrality gap.

The Greedy-Lagrangian approach achieves near-optimal performance in finite samples.

Abstract

We study optimal policy learning under combined budget and minimum coverage constraints. We show that the problem admits a knapsack-type structure and that the optimal policy can be characterized by an affine threshold rule involving both budget and coverage shadow prices. We establish that the linear programming relaxation of the combinatorial solution has an O(1) integrality gap, implying asymptotic equivalence with the optimal discrete allocation. Building on this result, we analyze two implementable approaches: a Greedy-Lagrangian (GLC) and a rank-and-cut (RC) algorithm. We show that the GLC closely approximates the optimal solution and achieves near-optimal performance in finite samples. By contrast, RC is approximately optimal whenever the coverage constraint is slack or costs are homogeneous, while misallocation arises only when cost heterogeneity interacts with a binding coverage constraint. Monte Carlo evidence supports these findings.