Inference on the value of a linear program

This paper studies inference on the value of a linear program (LP) when both the objective function and constraints are possibly unknown and must be estimated from data. We show that many inference problems in partially identified models can be reformulated in this way. Building on Shapiro (1991) and Fang and Santos (2019), we develop a pointwise valid inference procedure for the value of an LP. We modify this pointwise inference procedure to construct one-sided inference procedures that are uniformly valid over large classes of data-generating processes. Our results provide alternative testing procedures for problems considered in Andrews et al. (2023), Cox and Shi (2023), and Fang et al. (2023) (in the low-dimensional case), and remain valid when key components--such as the coefficient matrix--are unknown and must be estimated. Moreover, our framework also accommodates inference on the identified set of a subvector, in models defined by linear moment inequalities, and does so under weaker constraint qualifications than those in Gafarov (2025).