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Barrier Functions and Interior-Point Algorithms for Linear Programming with Zero-, One-, or Two-Sided Bounds on the Variables

Barrier Functions and Interior-Point Algorithms for Linear Programming with Zero-, One-, or... This study examines two different barrier functions and their use in both path-following and potential-reduction interior-point algorithms for solving a linear program of the form: minimize cTx subject to Ax b and l x u, where components of l and u can be nonfinite, so the variables xj, can have 0-, 1-, or 2-sided bounds, j 1, , n. The barrier functions that we study include an extension of the standard logarithmic barrier function and an extension of a barrier function introduced by Nesterov. In the case when both l and u have all of then components finite, these barrier functions are$$\Psi(x)=\sum_j\left\{-\ln(u_j - x_j) -\ln (x_j - l_j)\right\}$$and$$\Psi(x)=\sum_j\left\{-\ln\left(\min\{u_j - x_j, x_j - l_j\}\right)+\min \{u_j - x_j, x_j - l_j\} / ((u_j - l_j)/2)\right\}.$$Each of these barrier functions gives rise to suitable primal and dual metrics that are used to develop both path-following and potential-reduction interior-point algorithms for solving such linear programming problems. The resulting complexity bounds on the algorithms depend only on the number of bounded variables, rather than on the number of finite inequalities in the system 1 x u, in contrast to the standard complexity bounds for interior-point algorithms. These enhanced complexity bounds stem directly from the choice of a natural metric induced by the barrier function. This study also demonstrates the inter-connection between the notion of self-concordance (introduced by Nesterov and Nemirovsky) and properties of the two barrier functions that drive the results contained herein. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematics of Operations Research INFORMS

Barrier Functions and Interior-Point Algorithms for Linear Programming with Zero-, One-, or Two-Sided Bounds on the Variables

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Publisher
INFORMS
Copyright
Copyright © INFORMS
Subject
Research Article
ISSN
0364-765X
eISSN
1526-5471
DOI
10.1287/moor.20.2.415
Publisher site
See Article on Publisher Site

Abstract

This study examines two different barrier functions and their use in both path-following and potential-reduction interior-point algorithms for solving a linear program of the form: minimize cTx subject to Ax b and l x u, where components of l and u can be nonfinite, so the variables xj, can have 0-, 1-, or 2-sided bounds, j 1, , n. The barrier functions that we study include an extension of the standard logarithmic barrier function and an extension of a barrier function introduced by Nesterov. In the case when both l and u have all of then components finite, these barrier functions are$$\Psi(x)=\sum_j\left\{-\ln(u_j - x_j) -\ln (x_j - l_j)\right\}$$and$$\Psi(x)=\sum_j\left\{-\ln\left(\min\{u_j - x_j, x_j - l_j\}\right)+\min \{u_j - x_j, x_j - l_j\} / ((u_j - l_j)/2)\right\}.$$Each of these barrier functions gives rise to suitable primal and dual metrics that are used to develop both path-following and potential-reduction interior-point algorithms for solving such linear programming problems. The resulting complexity bounds on the algorithms depend only on the number of bounded variables, rather than on the number of finite inequalities in the system 1 x u, in contrast to the standard complexity bounds for interior-point algorithms. These enhanced complexity bounds stem directly from the choice of a natural metric induced by the barrier function. This study also demonstrates the inter-connection between the notion of self-concordance (introduced by Nesterov and Nemirovsky) and properties of the two barrier functions that drive the results contained herein.

Journal

Mathematics of Operations Research – INFORMS

Published: May 1, 1995

Keywords: Keywords : linear programming ; barrier functions ; interior-point algorithms ; complexity theory ; self-concordance

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