In this paper, we present a global optimization method based on the filled function method to solve systems of nonlinear equations. Formulating a system of nonlinear equation into an equivalent global optimization problem, we manage to find a solution or an appropriate solution of the system of nonlinear equations by solving the formulated global optimization problem. A novel filled function method is proposed to solve the global optimization problem. Two numerical examples are presented to illustrate the efficiency of this method.
Close form analytical techniques for the design of a certain class of recursive digital filters such as the elliptic filter have appeared. Such close form analytical techniques are suitable for designing filters with piece-wise constant magnitude response. The design of recursive digital filters with arbitrary frequency response is a nonlinear optimization problem. Specifically, it belongs to the class of global bi-lever programming problem. Optimal solution for a global bi-lever programming problem is notoriously difficult to obtain. In this paper, the bi-lever programming problem is first converted into a differentiate one-lever problem. Consequently we not only prove that the global minimizer of the converted one-lever problem is an approximate global minimizer of the original bi-lever problem, but also a novel filled function method for the design of recursive digital filters meeting arbitrary frequency response specifications is proposed. Several design examples are presented to illustrate our new technique