Extensions of a result on the synthesis of signals in the presence of inconsistent constraints

Abstract

Image restoration in the presence of compatible convex constraints can be carried out by the method of convex projections [1]-[3]. In a recent interesting paper [4], Goldburg and Marks have used a modified version of the above technique to solve an optimization problem involving the synthesis of a signal subject to two inconsistent constraints. We complete this result and also show that their restriction to a real Hilbert space setting is unnecessary. A unique generalization of the above optimization problem to the case of more than two constraints does not seem possible. Nevertheless, considerations of symmetry have led us to a formulation which identifies minimizers as nodes on closed greedy paths and an important and potentially useful property of such paths is proven in Theorem 4.