A convex optimization approach for solving large scale linear systems
The well-known Conjugate Gradient (CG) method minimizes a strictly convex quadratic function for solving large-scale linear system of equations when the coefficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-...
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| Format: | Article |
| Language: | English |
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Universidad Simón Bolívar
2016-11-01
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| Series: | Bulletin of Computational Applied Mathematics |
| Subjects: | |
| Online Access: | http://drive.google.com/open?id=0B5GyVVQ6O030Z2pHbWx0cHhyTnc |
| _version_ | 1818138278810877952 |
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| author | Debora Cores Johanna Figueroa |
| author_facet | Debora Cores Johanna Figueroa |
| author_sort | Debora Cores |
| collection | DOAJ |
| description | The well-known Conjugate Gradient (CG) method minimizes a strictly convex quadratic function for solving large-scale linear system of equations when the coefficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-scale linear system of equations regardless of the characteristics of the coefficient matrix. For finding the global minimizers, of this new convex function, any low-cost iterative optimization technique could be applied. In particular, we propose to use the low-cost globally convergent Spectral Projected Gradient (SPG) method, which allow us to extend this optimization approach for solving consistent square and rectangular linear system, as well as linear feasibility problem, with and without convex constraints and with and without preconditioning strategies. Our numerical results indicate that the new scheme outperforms state-of-the-art iterative techniques for solving linear systems when the symmetric part of the coefficient matrix is indefinite, and also for solving linear feasibility problems. |
| first_indexed | 2024-12-11T10:09:40Z |
| format | Article |
| id | doaj.art-efd48df9a94043e2abb6ddcf60473d40 |
| institution | Directory Open Access Journal |
| issn | 2244-8659 2244-8659 |
| language | English |
| last_indexed | 2024-12-11T10:09:40Z |
| publishDate | 2016-11-01 |
| publisher | Universidad Simón Bolívar |
| record_format | Article |
| series | Bulletin of Computational Applied Mathematics |
| spelling | doaj.art-efd48df9a94043e2abb6ddcf60473d402022-12-22T01:11:47ZengUniversidad Simón BolívarBulletin of Computational Applied Mathematics2244-86592244-86592016-11-01515376A convex optimization approach for solving large scale linear systemsDebora Cores0Johanna Figueroa1Departamento de Cómputo Científico y Estadística, Universidad Simón Bolívar (USB), Caracas 1080-A, VenezuelaDepartamento de Matemática de la Facultad de Ciencias y Tecnología, Universidad de Carabobo (UC), Valencia 2005, VenezuelaThe well-known Conjugate Gradient (CG) method minimizes a strictly convex quadratic function for solving large-scale linear system of equations when the coefficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-scale linear system of equations regardless of the characteristics of the coefficient matrix. For finding the global minimizers, of this new convex function, any low-cost iterative optimization technique could be applied. In particular, we propose to use the low-cost globally convergent Spectral Projected Gradient (SPG) method, which allow us to extend this optimization approach for solving consistent square and rectangular linear system, as well as linear feasibility problem, with and without convex constraints and with and without preconditioning strategies. Our numerical results indicate that the new scheme outperforms state-of-the-art iterative techniques for solving linear systems when the symmetric part of the coefficient matrix is indefinite, and also for solving linear feasibility problems.http://drive.google.com/open?id=0B5GyVVQ6O030Z2pHbWx0cHhyTncNonlinear convex optimizationspectral gradient methodlarge-scale linear systems |
| spellingShingle | Debora Cores Johanna Figueroa A convex optimization approach for solving large scale linear systems Bulletin of Computational Applied Mathematics Nonlinear convex optimization spectral gradient method large-scale linear systems |
| title | A convex optimization approach for solving large scale linear systems |
| title_full | A convex optimization approach for solving large scale linear systems |
| title_fullStr | A convex optimization approach for solving large scale linear systems |
| title_full_unstemmed | A convex optimization approach for solving large scale linear systems |
| title_short | A convex optimization approach for solving large scale linear systems |
| title_sort | convex optimization approach for solving large scale linear systems |
| topic | Nonlinear convex optimization spectral gradient method large-scale linear systems |
| url | http://drive.google.com/open?id=0B5GyVVQ6O030Z2pHbWx0cHhyTnc |
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