Newton’s method and its use in optimization

Newton’s method is a basic tool in numerical analysis and numerous applications, including operations research and data mining. We survey the history of the method, its main ideas, convergence results, modifications, its global behavior. We focus on applications of the method for various classes of optimization problems, such as unconstrained minimization, equality constrained problems, convex programming and interior point methods. Some extensions (non-smooth problems, continuous analog, Smale’s results, etc.) are discussed briefly, while some others (e.g., versions of the method to achieve global convergence) are addressed in more details.