Adapting Broyden method to handle linear constraints imposed via Lagrange multipliers

Abstract

Various non-linear equation solvers are adapted to handle linear constraints via the Lagrange-multiplier technique. This adaptation process turns out to be quite straightforward for Newton-Raphson methods and rank-two Quasi-Newton methods (BFGS and DFP), but rather more involved for Broyden method. In fact, two Broyden methods can be obtained: the standard one and a modified one, better adapted to the Lagrange-multiplier environment. Some numerical examples are used to assess the relative performance of the various adapted solvers. These tests illustrate the superiority of the modified Broyden method over the standard one.