Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications
In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the <i>cover order</i> and the <i>cover length</i>, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the ma...
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/10/2338 |
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| author | Xiaoxuan Chu Kon Max Wong Jun Chen Jiankang Zhang |
| author_facet | Xiaoxuan Chu Kon Max Wong Jun Chen Jiankang Zhang |
| author_sort | Xiaoxuan Chu |
| collection | DOAJ |
| description | In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the <i>cover order</i> and the <i>cover length</i>, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with <i>non-negativity constraints</i> on variables for both <i>homogeneous</i> and <i>nonhomogeneous</i> cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including <i>linear programming</i> (LP) problems and <i>non-negative least squares</i> (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem. |
| first_indexed | 2024-03-11T03:30:58Z |
| format | Article |
| id | doaj.art-1531b67d4e2a47daa7f8f155ead9ea64 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T03:30:58Z |
| publishDate | 2023-05-01 |
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| spelling | doaj.art-1531b67d4e2a47daa7f8f155ead9ea642023-11-18T02:19:33ZengMDPI AGMathematics2227-73902023-05-011110233810.3390/math11102338Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its ApplicationsXiaoxuan Chu0Kon Max Wong1Jun Chen2Jiankang Zhang3Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, CanadaDepartment of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, CanadaDepartment of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, CanadaDepartment of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L8, CanadaIn this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the <i>cover order</i> and the <i>cover length</i>, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with <i>non-negativity constraints</i> on variables for both <i>homogeneous</i> and <i>nonhomogeneous</i> cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including <i>linear programming</i> (LP) problems and <i>non-negative least squares</i> (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem.https://www.mdpi.com/2227-7390/11/10/2338hyper-rectangle covercover ordercover lengthsystem of linear equationsnon-negativity constraintsnon-negative least squares |
| spellingShingle | Xiaoxuan Chu Kon Max Wong Jun Chen Jiankang Zhang Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications Mathematics hyper-rectangle cover cover order cover length system of linear equations non-negativity constraints non-negative least squares |
| title | Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications |
| title_full | Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications |
| title_fullStr | Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications |
| title_full_unstemmed | Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications |
| title_short | Systems of Linear Equations with Non-Negativity Constraints: Hyper-Rectangle Cover Theory and Its Applications |
| title_sort | systems of linear equations with non negativity constraints hyper rectangle cover theory and its applications |
| topic | hyper-rectangle cover cover order cover length system of linear equations non-negativity constraints non-negative least squares |
| url | https://www.mdpi.com/2227-7390/11/10/2338 |
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