COMBINED LP AND QUASI-NEWTON METHODS FOR NONLINEAR L1 OPTIMIZATION

This paper investigates the theoretical properties of an algorithm for minimizing the sum of absolute values of a set of nonlinear functions. The algorithm is a combination of a first order method that approximates the solution by successive linear programming, and a quasi-Newton method using approximate second order information to solve the system of nonlinear equations arising from the necessary first order conditions at a solution. The latter method is intended to be used only in the final stages of the iteration, but several switches between the two methods may take place. It is proved that under mild conditions the combined method can converge only to stationary points. The final convergence is quadratic in case of a strict local minimum.