Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when the barrier parameter is reduced at a reasonable rate. We present local analysis showing why a pure Newton step in a long-step barrier method for nonlinearly constrained optimization may be seriously infeasible, even when taken from an apparently favorable point. The features described are illustrated numerically and connected to known theoretical results for convex problems satisfying self-concordancy assumptions. We also indicate the contrasting nature of an approximate step to the desired minimizer of the barrier function. 1. Introduction 1.1. Background Interior methods, most commonly based on barrier functions, have been applied with great practical success in recent years to many con...

Barrier methods for constrained optimization, widely applied in the 1960's and 1970's, have recently enjoyed a revival of popularity. For problems involving nonlinear functions, some modern interior methods include a line search with respect to a logarithmic barrier function or related potential function. Standard line search procedures tend to be inefficient in this context for two reasons: an inappropriate choice of initial trial step, and poor approximation of the barrier function by low-order polynomial interpolants. This paper discusses line search strategies specifically designed for the logarithmic barrier function. 1. Barrier Methods for Constrained Optimization Beginning in 1984 with the work of Karmarkar [Kar84], a resurgence of interest has taken place in barrier methods for constrained optimization. Classical barrier methods, which were popular in the 1960's and early 1970's, treat inequality constraints by creating a transformed function containing a positive singularity ...

In the recent past a number of papers were written that present low complexity interior-point methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is self-concordant, and that the analyses of interiorpoint methods for these programs can thus be reduced to the analysis of interior-point methods with self-concordant barrier functions. Key words: interior-point method, barrier function, dual geometric programming, (extended) entropy programming, primal and dual l p -programming, relative Lipschitz condition, scaled Lipschitz condition, self-concordance. 1 Introduction The efficiency of a barrier method for solving convex programs strongly depends on the properties of the barrier function used. A key property that is sufficient to prove fast convergence for barrier methods is the property of self-concordance introduced in [17]. This condition not only allows a proof of polynomial convergen...