This paper studies a regularized support function estimator for bounds on components of the parameter vector in the case in which the identified set is a polygon. The proposed regularized estimator has three important properties: (i) it has a uniform asymptotic Gaussian limit in the presence of flat faces in the absence of redundant (or overidentifying) constraints (or vice versa); (ii) the bias from regularization does not enter the first-order limiting distribution; (iii) the estimator remains consistent for sharp (non-enlarged) identified set for the individual components even in the non-regualar case. These properties are used to construct \emph{uniformly valid} confidence sets for an element $θ_{1}$ of a parameter vector $θ\in\mathbb{R}^{d}$ that is partially identified by affine moment equality and inequality conditions. The proposed confidence sets can be computed as a solution to a small number of linear and convex quadratic programs, leading to a substantial decrease in computation time and guarantees a global optimum. As a result, the method provides a uniformly valid inference in applications in which the dimension of the parameter space, $d$, and the number of inequalities, $k$, were previously computationally unfeasible ($d,k=100$). The proposed approach can be extended to construct confidence sets for intersection bounds, to construct joint polygon-shaped confidence sets for multiple components of $θ$, and to find the set of solutions to a linear program. Inference for coefficients in the linear IV regression model with an interval outcome is used as an illustrative example.

The earlier version of the paper was previously circulated under title "Inference on scalar parameters in set-identified affine models" and was a chapter in my PhD dissertation