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Why a Pure Primal Newton Barrier Step May Be Infeasible
 SIAM Journal on Optimization
, 1993
"... 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 ..."
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Cited by 21 (3 self)
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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 longstep 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 selfconcordancy 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...
Line Search Procedures for the Logarithmic Barrier Function
 SIAM Journal on Optimization
, 1991
"... 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 fun ..."
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Cited by 17 (5 self)
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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 loworder 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 ...
A Unifying Investigation of InteriorPoint Methods for Convex Programming
 Faculty of Mathematics and Informatics, TU Delft, NL2628 BL
, 1992
"... In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interiorpo ..."
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Cited by 5 (4 self)
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In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interiorpoint methods for these programs can thus be reduced to the analysis of interiorpoint methods with selfconcordant barrier functions. Key words: interiorpoint method, barrier function, dual geometric programming, (extended) entropy programming, primal and dual l p programming, relative Lipschitz condition, scaled Lipschitz condition, selfconcordance. 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 selfconcordance introduced in [17]. This condition not only allows a proof of polynomial convergen...