Barrier-newton methods in mathematical programming

Abstract

A space transformation technique is used for the reduction of constrained minimization problems to minimization problems without inequality constraints. The continuous and discrete versions of Newton's method are applied for solving such reduced LP and NLP problems. The space transformation modifies these methods and introduces additional matrices which play the role of a multiplicative barrier, preventing the trajectories from crossing the boundary of the feasible set. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equations. These algorithms do not require feasibility of starting and current points, but they preserve feasibility. The discrete version of the barrier-Newton method has a superlinear convergence rate and a special stepsize regulation gives quadratic convergence.