A Dual Approach to Linear Inverse Problems with Convex Constraints

Abstract

A simple constraint qualification is developed and used to derive an explicit solution to a constrained optimization problem in Hilbert space. A finite parameterization is obtained for the minimum norm element in the intersection of a linear variety of finite co-dimension and a closed convex constraint set. The result extends previous duality theorems for convex cone set constraints. A fixed point iteration is presented for computing the parameters and yields a least-squares solution when the variety and constraint set have empty intersection. Proofs rely on nearest-point projections onto convex sets and the properties of monotone, firmly nonexpansive, and averaged mappings.