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Volume 42, Issue 4
A Low-Cost Optimization Approach for Solving Minimum Norm Linear Systems and Linear Least-Squares Problems

Debora Cores & Johanna Figueroa

J. Comp. Math., 42 (2024), pp. 932-954.

Published online: 2024-04

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  • Abstract

Recently, the authors proposed a low-cost approach, named Optimization Approach for Linear Systems (OPALS) for solving any kind of a consistent linear system regarding the structure, characteristics, and dimension of the coefficient matrix $A.$ The results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known techniques. In this work, we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem (LLSP) and the Minimum Norm Linear System Problem (MNLSP) using any iterative low-cost gradient-type method, avoiding the construction of the matrices $A^TA$ or $AA^T,$ and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction. The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both problems. Moreover, the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems. We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.

  • AMS Subject Headings

65F10, 65F20, 90C06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-932, author = {Cores , Debora and Figueroa , Johanna}, title = {A Low-Cost Optimization Approach for Solving Minimum Norm Linear Systems and Linear Least-Squares Problems}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {4}, pages = {932--954}, abstract = {

Recently, the authors proposed a low-cost approach, named Optimization Approach for Linear Systems (OPALS) for solving any kind of a consistent linear system regarding the structure, characteristics, and dimension of the coefficient matrix $A.$ The results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known techniques. In this work, we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem (LLSP) and the Minimum Norm Linear System Problem (MNLSP) using any iterative low-cost gradient-type method, avoiding the construction of the matrices $A^TA$ or $AA^T,$ and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction. The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both problems. Moreover, the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems. We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2301-m2021-0313}, url = {http://global-sci.org/intro/article_detail/jcm/23041.html} }
TY - JOUR T1 - A Low-Cost Optimization Approach for Solving Minimum Norm Linear Systems and Linear Least-Squares Problems AU - Cores , Debora AU - Figueroa , Johanna JO - Journal of Computational Mathematics VL - 4 SP - 932 EP - 954 PY - 2024 DA - 2024/04 SN - 42 DO - http://doi.org/10.4208/jcm.2301-m2021-0313 UR - https://global-sci.org/intro/article_detail/jcm/23041.html KW - Nonlinear convex optimization, Gradient-type methods, Spectral gradient method, Minimum norm solution linear systems, Linear least-squares solution. AB -

Recently, the authors proposed a low-cost approach, named Optimization Approach for Linear Systems (OPALS) for solving any kind of a consistent linear system regarding the structure, characteristics, and dimension of the coefficient matrix $A.$ The results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known techniques. In this work, we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem (LLSP) and the Minimum Norm Linear System Problem (MNLSP) using any iterative low-cost gradient-type method, avoiding the construction of the matrices $A^TA$ or $AA^T,$ and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction. The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both problems. Moreover, the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems. We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.

Cores , Debora and Figueroa , Johanna. (2024). A Low-Cost Optimization Approach for Solving Minimum Norm Linear Systems and Linear Least-Squares Problems. Journal of Computational Mathematics. 42 (4). 932-954. doi:10.4208/jcm.2301-m2021-0313
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