🤖 AI Summary
This study addresses the problem of learning optimal decision policies under dual constraints of a budget limit and a minimum coverage requirement, formulating it for the first time as a combinatorial optimization problem with a knapsack structure. Through linear programming relaxation and Lagrangian duality analysis, the authors establish that the problem exhibits a constant-order integrality gap and derive an affine threshold rule—based on shadow prices—that characterizes the optimal policy. Two efficient algorithms are proposed: the Greedy-Lagrangian algorithm achieves near-optimal performance with finite samples, while the rank-and-cut algorithm is provably near-optimal when costs are homogeneous or the coverage constraint is loose, exhibiting only minor misallocations in the challenging regime of heterogeneous costs and tight coverage constraints.
📝 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.