Solving function approximation problems using the L2-Norm of the log ratio as a metric

Abstract

This article considers the following function approximation problem: Given a non-negative function and a set of equality constraints, find the closest to it non-negative function which satisfies the constraints. As a measure of distance we propose the L2-norm of the logarithm of the ratio of the two functions. As shown, this metric guarantees that (i) the sought function is non-negative and (ii) to the extent to which the constraints allow, the magnitude of the difference between the sought and the given function is proportional to the magnitude of the given function. To solve the problem we convert it to a finite dimensional constrained optimization problem and apply the method of Lagrange multipliers. The resulting nonlinear system, together with the system for the constraints, are solved self-consistently by applying an appropriate iterative procedure.