Inference on Linear Programs via Feasibility Testing

This paper studies inference on the value of a linear program (LP) when the objective and constraints are estimated. Using LP duality, we construct one- and two-sided confidence intervals by inverting tests of the feasibility of systems of linear inequalities. Existing tests yield asymptotically valid confidence intervals but can have poor power. We propose a modification that augments these systems with estimated ``box constraints.'' The resulting procedure has power against $N^{-1/2}$-local alternatives and yields confidence intervals of order $N^{-1/2}$. These results require only that the LP value be finite, and do not require compactness of the feasible region nor standard constraint qualifications. We obtain stronger guarantees under various constraint qualifications and establish uniform validity over large classes of DGPs when known deterministic constraints bound the primal and/or dual variables. Simulations demonstrate good finite-sample performance.