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 cover order and the cover length, 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 non-negativity constraints on variables for both homogeneous and nonhomogeneous 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 linear programming (LP) problems and non-negative least squares (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.