Centered newton method for mathematical programming

Abstract

The purpose of this paper is to introduce a generic class of algorithms for solving a system of nonlinear equations, Linear Programming problems, Quadratic Programming problems, Nonlinear Programming problems and general complementarity problems. The algorithms were obtained by modifying the standard Newton-Raphson method applied to a system of nonlinear equations in the complementarity conditions so that it is biased towards ‘center curve’ passing through the solutions. The search direction of the methods is a positive combination of the Newton direction and a ‘centering’ direction which is also given by applying the Newton method to a projected system of the complementarity equations. These two directions guide the generated sequence of the approximations towards the solution and the center variety respectively. A class of ‘penalized norms’ and ‘guiding cones’ is also introduced for choosing step lengths in bivariate search.