Comment: 49 pages (110 with appendix), 12 figures (16 with appendix)Sharp bounds on partially identified parameters are often given by the values of linear programs (LPs). This paper introduces a novel estimator of the LP value. Unlike existing procedures, our estimator is root-n-consistent, pointwise in the probability measure, whenever the population LP is feasible and finite. Our estimator is valid under point-identification, over-identifying constraints, and solution multiplicity. Turning to uniformity properties, we prove that the LP value cannot be uniformly consistently estimated without restricting the set of possible distributions. We then show that our estimator achieves uniform consistency under a condition that is minimal for the existence of any such estimator. We obtain computationally efficient, asymptotically normal inference procedure with exact asymptotic coverage at any fixed probability measure. To complement our estimation results, we derive LP sharp bounds in a general identification setting. We apply our findings to estimating returns to education. To that end, we propose the conditionally monotone IV assumption (cMIV) that tightens the classical monotone IV (MIV) bounds and is testable under a mild regularity condition. Under cMIV, university education in Colombia is shown to increase the average wage by at least $5.5\%$, whereas classical conditions fail to yield an informative bound.