Homotopy continuation method in avoiding the problem of divergence of traditional newton's method

Abstract

The traditional Newton’s Method is known as a popular method for solving optimization problem of non-linear functions. It is derived from the efficiency in the convergence speed. However, Newton’s Method usually will yield divergence especially when the initial value is far away from the exact solution. In another situation, divergence also occur when the second derivative in the numerical iteration formula of Newton’s Method is equal to zero or tends to zero. Homotopy Continuation Method has the ability to overcome this problem. The purpose of this research is to probe the step taken in Homotopy Continuation Method in avoiding the problem of divergence in Traditional Newton’s Method. Homotopy Continuation Method is a kind of perturbation method that can guarantee the answer by a certain path if we choose the auxiliary homotopy function. This method transforms a complicated situation into a simpler one that is easy to solve and gradually deform the simpler problem into the original one by computing the extremizers of the intervening problems and eventually ending with an extremum of the original problem. To strengthen the findings, this thesis presents a description of a MATLAB code that implements the Homotopy Continuation Method and Newton’s Method for solving the optimization problem. This study succeeded in avoiding the problem of divergence of traditional Newton Method and can guarantee the answer.