Abstract
The Gaussian elimination method is usually used in solving problems related to systems of linear inequalities. The present review paper describes the application of the least-squares method to studying problems connected with linear inequalities (like redundant inequalities, theorems of alternative, mathematical programming). The minimum norm solution to the system of linear inequalities is found by solving a non-negative least-squares (NNLS) problem. A linear programming (LP) problem is transformed to the system of inequalities in several ways. By solving the corresponding NNLS problem an initial solution to the LP problem is found. The main ideas are explained by simple examples.
Highlights
The Gaussian elimination method is usually used in solving problems related to systems of linear inequalities
The present review paper describes the application of the least-squares method to studying problems connected with linear inequalities
The minimum norm solution to the system of linear inequalities is found by solving a non-negative least-squares (NNLS) problem
Summary
Linear inequalities are more complicated than linear equations, because equations as constraints can be transformed into linear inequalities by replacing each equation with the opposing pair of inequalities. The elimination method for solving linear equations remained unknown in Europe until K. Gauss rediscovered it in the 19th century. Until recently there was no computationally viable method for solving systems of linear constraints including inequalities. The present paper deals with the solution of a system of linear inequalities and other problems by applying the least-squares method (see Section 4). This highly developed method is much older than the simplex method. In this paper it will be proved that such an idea can be used for solving systems of linear inequalities and mathematical programming problems
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