Local behavior of the Newton method on two equivalent systems from linear programming

Authors:
C. Villalobos;R. Tapia;Y. Zhang
Affiliations:
Assistant Professor, Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas;Professor, Department of Computational and Applied Mathematics, Rice University, Houston, Texas;Associate Professor, Department of Computational and Applied Mathematics, Rice University, Houston, Texas
Venue:
Journal of Optimization Theory and Applications
Year:
2002

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Abstract

Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce significantly different behavior when applied to equivalent systems, i.e., problems with the same solution but different mathematical formulations. In this paper, we investigate differences in the local behavior of Newton's method when applied to two different but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier function formulation and the equations in the so-called perturbed optimality conditions. Through theoretical analysis and numerical results, we provide an explanation of why Newton's method performs more effectively on the latter system.