# A State of the Art Review of Systems of Linear Inequalities and Related Observability Problems
## 1 Introduction and Motivation  
Linear systems of inequalities appear very frequently and play a central role in real practice. Cases of special relevance are those of linear programming problems, but there are many more. However, in spite of their importance, the way of dealing with these systems of equations is not well known and has not reached sufficiently the group of people working with these problems and their applications. In addition, the software available to solve them is not easily available and most people ignore where to find it. Consequently, this field has not advanced as expected. It is frustrating to enter the internet and look for methods to solve these systems and, incomprehensibly, find lots of entries in which only the cases of one, two or, at most, three inequalities are dealt with, giving the impression that general methods for large systems do not exist, when this is not the case. Some available software for small problems can be downloaded from: ([[https://meteo.unican.es/temp/castie/GammaAlgorithm/GammaAlgorithm.zip](https://meteo.unican.es/temp/castie/GammaAlgorithm/GammaAlgorithm.zip)]([https://meteo.unican.es/temp/castie/GammaAlgorithm/GammaAlgorithm.zip](https://meteo.unican.es/temp/castie/GammaAlgorithm/GammaAlgorithm.zip)), accessed on 12 July 2023). In addition, the problem of the solution of these systems has not been connected sufficiently with their associated problems in other fields, in which important advances were produced a one time ago. This is the case of identifying the main components of polyhedra, such as edges, vertices, and facets, together with the methods to obtain thennumerically (see [1], who finds the convex hull, facet by facet, and [2], who provide a pivoting algorithm to obtain facets and vertices). Since most linear systems of inequalities arising in real practice problems are bounded, that is, their sets of solutions are polytopes generated by their vertices, and methods for finding the vertices of a polytope are well known (e.g., [2,3] or [4]), they should be used very frequently to solve them. Unfortunately, this is not the case. If one looks at the less frequent case of unbounded problems, similar conclusions can be drawn. Due to its connection with linear programming, some methods to solve systems of linear inequalities have been based on this problem, as, for example, the one described in [5], which solves only some particular cases. In this paper, concerned with this fact and trying to break with this undesirable situation, we deal with a summary of the state of the art of these systems of inequalities and review some of the existing work, including some illustrative examples to motivate users and researchers to use these relevant tools and to facilitate and make available these important tools to people. The paper is organized as follows. In Section 2, linear systems of equations are discussed with special consideration to their compatibility conditions and the identification of the subsets of unknowns that are unbounded or have unique solutions. In Section 3, we analyze systems of linear inequalities, homogeneous and complete, together with their corresponding compatibility problems, which become more complex than the case of systems of linear equations, because here dual cones replace orthogonal subspaces. In Section 4, the observability problem is presented and analyzed. We will see that between the two extremes of incompatibility and whole observability there is an interesting range of partial observability cases, which have important applications. In systems of linear inequalities in which there are infinitely many solutions, some set of unknowns could be uniquely identified, and for those with multiple solutions, lower and upper bounds and several other interesting items of information can be easily obtained. In Section 5, we demonstrate that the problem of linear programming becomes trivial if the system of constraints is initially solved or stated directly in its polyhedral form.