Computing Weighted Analytic Center for Linear Matrix Inequalities Using Infeasible Newton's Method

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Abstract

We study the problem of computing weighted analytic center for system of linear matrix inequality constraints. The problem can be solved using Standard Newton's method. However, this approach requires that a starting point in the interior point of the feasible region be given or a Phase I problem be solved. We address the problem by using Infeasible Newton's method applied to the KKT system of equations which can be started from any point. We implement the method using backtracking line search technique and also study the effect of large weights on the method. We use numerical experiments to compare Infeasible Newton's method with Standard Newton's method. The results show that Infeasible Newton's method moves in the interior of the feasible regions often very quickly, starting from any point. We recommend it as a method for finding an interior point by setting each weight to be 1. It appears to work better than Standard Newton's method in finding the weighted analytic center when none of weights is very large relative to the other weights. However, we find that Infeasible Newton's method is more sensitive than Standard Newton's method to large variation in the weights.

Original languageEnglish (US)
Article number456392
JournalJournal of Mathematics
Volume2015
DOIs
StatePublished - 2015

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Analytic Center
Newton Methods
Matrix Inequality
Linear Inequalities
Computing
Feasible region
Interior Point
KKT System
Backtracking
Line Search
Inequality Constraints
System of equations
Interior
Numerical Experiment
Standards

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Computing Weighted Analytic Center for Linear Matrix Inequalities Using Infeasible Newton's Method",
abstract = "We study the problem of computing weighted analytic center for system of linear matrix inequality constraints. The problem can be solved using Standard Newton's method. However, this approach requires that a starting point in the interior point of the feasible region be given or a Phase I problem be solved. We address the problem by using Infeasible Newton's method applied to the KKT system of equations which can be started from any point. We implement the method using backtracking line search technique and also study the effect of large weights on the method. We use numerical experiments to compare Infeasible Newton's method with Standard Newton's method. The results show that Infeasible Newton's method moves in the interior of the feasible regions often very quickly, starting from any point. We recommend it as a method for finding an interior point by setting each weight to be 1. It appears to work better than Standard Newton's method in finding the weighted analytic center when none of weights is very large relative to the other weights. However, we find that Infeasible Newton's method is more sensitive than Standard Newton's method to large variation in the weights.",
author = "Shafiu Jibrin",
year = "2015",
doi = "10.1155/2015/456392",
language = "English (US)",
volume = "2015",
journal = "Journal of Mathematics",
issn = "2314-4629",
publisher = "Hindawi Publishing Corporation",

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