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Volume 10, Issue 3
An Algorithm for Finding Nonnegative Minimal Norm Solutions of Linear Systems

S. Bahi & A. Ross

Int. J. Numer. Anal. Mod., 10 (2013), pp. 745-755.

Published online: 2013-10

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  • Abstract

A system of linear equations $Ax = b$, in $n$ unknowns and $m$ equations which has a nonnegative solution is considered. Among all its solutions, the one which has the least norm is sought when $\mathbb{R}^n$ is equipped with a strictly convex norm. We present a globally convergent, iterative algorithm for computing this solution. This algorithm takes into account the special structure of the problem. Each iteration cycle of the algorithm involves the solution of a similar quadratic problem with a modified objective function. Duality conditions for optimality are studied. Feasibility and global convergence of the algorithm are proved. As a special case we implemented and tested the algorithm for the $\ell^p$-norm, where $1 < p < ∞$. Numerical results are included.

  • Keywords

Linear equations, Least norms, Optimality, Duality conditions.

  • AMS Subject Headings

90C25, 90C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-745, author = {S. Bahi and A. Ross}, title = {An Algorithm for Finding Nonnegative Minimal Norm Solutions of Linear Systems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {3}, pages = {745--755}, abstract = {

A system of linear equations $Ax = b$, in $n$ unknowns and $m$ equations which has a nonnegative solution is considered. Among all its solutions, the one which has the least norm is sought when $\mathbb{R}^n$ is equipped with a strictly convex norm. We present a globally convergent, iterative algorithm for computing this solution. This algorithm takes into account the special structure of the problem. Each iteration cycle of the algorithm involves the solution of a similar quadratic problem with a modified objective function. Duality conditions for optimality are studied. Feasibility and global convergence of the algorithm are proved. As a special case we implemented and tested the algorithm for the $\ell^p$-norm, where $1 < p < ∞$. Numerical results are included.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/593.html} }
TY - JOUR T1 - An Algorithm for Finding Nonnegative Minimal Norm Solutions of Linear Systems AU - S. Bahi & A. Ross JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 745 EP - 755 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/593.html KW - Linear equations, Least norms, Optimality, Duality conditions. AB -

A system of linear equations $Ax = b$, in $n$ unknowns and $m$ equations which has a nonnegative solution is considered. Among all its solutions, the one which has the least norm is sought when $\mathbb{R}^n$ is equipped with a strictly convex norm. We present a globally convergent, iterative algorithm for computing this solution. This algorithm takes into account the special structure of the problem. Each iteration cycle of the algorithm involves the solution of a similar quadratic problem with a modified objective function. Duality conditions for optimality are studied. Feasibility and global convergence of the algorithm are proved. As a special case we implemented and tested the algorithm for the $\ell^p$-norm, where $1 < p < ∞$. Numerical results are included.

S. Bahi and A. Ross. (2013). An Algorithm for Finding Nonnegative Minimal Norm Solutions of Linear Systems. International Journal of Numerical Analysis and Modeling. 10 (3). 745-755. doi:
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