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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 = {}, 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 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 & A. Ross. (1970). 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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