Algorithmic solution for ℓ1-constrained projection in possibilistic IV posterior computation

Develop a complete and efficient computational procedure for the ℓ1-norm violation set A_τ = {α ∈ ℝ^p : ||α||_1 ≤ τ} within the possibilistic instrumental variable regression framework, specifically to compute for each β the projection α*(β) = argmin_{α : ||α||_1 ≤ τ} (α − t(β))^T (Z^T Z)(α − t(β)), where t(β) = ĥγ₁ − βĥγ₂, and integrate this projection into the evaluation of the conditional posterior possibility f(β | α ∈ A_τ, W). Provide algorithmic details—e.g., a LARS-type approach—and theoretical guarantees suitable for practical implementation.

Background

In the proposed possibilistic IV methodology, evaluating the conditional posterior possibility f(β | α ∈ A, W) requires solving a projection problem to find the α in a specified violation set A that minimizes the quadratic form (α − t(β))^T (Z^T Z)(α − t(β)). For rectangular sets A, this is handled via quadratic programming, and for the ℓ2-norm budget set A_τ = {α : ||α||_2 ≤ τ} a ridge-type closed-form solution is provided.

The authors note that an analogous treatment for an ℓ1-norm budget set would involve a LARS-type algorithm but do not provide the necessary algorithmic development. Completing this computational component is required to operationalize the method under ℓ1 constraints, which are natural in sparse or componentwise-bounded invalidity settings.