Optimal Policy Learning under Budget and Coverage Constraints
TLDR
This paper characterizes optimal policy learning under budget and coverage constraints, showing a knapsack structure and near-optimal algorithms.
Key contributions
- Characterizes optimal policy learning under budget and coverage constraints with a knapsack-type structure.
- Defines the optimal policy using an affine threshold rule based on budget and coverage shadow prices.
- Establishes an O(1) integrality gap for the LP relaxation, ensuring asymptotic equivalence with discrete allocation.
- Analyzes Greedy-Lagrangian (GLC) and rank-and-cut (RC) algorithms, showing GLC's near-optimal performance.
Why it matters
This paper provides a strong theoretical foundation for policy learning under common real-world constraints. Its insights into knapsack structure and integrality gaps offer new ways to approach resource allocation problems. The proposed algorithms, especially GLC, offer practical, near-optimal solutions for practitioners.
Original 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.
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