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This paper studies a regularized support function estimator for bounds oncomponents of the parameter vector in the case in which the identified set is apolygon. The proposed regularized estimator has three important properties: (i)it has a uniform asymptotic Gaussian limit in the presence of flat faces in theabsence of redundant (or overidentifying) constraints (or vice versa); (ii) thebias from regularization does not enter the first-order limitingdistribution;(iii) the estimator remains consistent for sharp identified setfor the individual components even in the non-regualar case. These propertiesare used to construct uniformly valid confidence sets for an element$\theta_{1}$ of a parameter vector $\theta\in\mathbb{R}^{d}$ that is partiallyidentified by affine moment equality and inequality conditions. The proposedconfidence sets can be computed as a solution to a small number of linear andconvex quadratic programs, which leads to a substantial decrease in computationtime and guarantees a global optimum. As a result, the method providesuniformly valid inference in applications in which the dimension of theparameter space, $d$, and the number of inequalities, $k$, were previouslycomputationally unfeasible ($d,k=100$). The proposed approach can be extendedto construct confidence sets for intersection bounds, to construct jointpolygon-shaped confidence sets for multiple components of $\theta$, and to findthe set of solutions to a linear program. Inference for coefficients in thelinear IV regression model with an interval outcome is used as an illustrativeexample.

Simple subvector inference on sharp identified set in affine models