# A State of the Art Review of Systems of Linear Inequalities and Related Observability Problems
## 3 Systems of Linear Inequalities
### Complete Systems of Linear Inequalities
#### 3.3.1 Example of Compatibility of the Particular Case of Complete System of Linear Inequalities
\begin{array}{rrrrrr}&+x_{2}&-x_{3}&-x_{4}&-b_{1}\\ +2x_{1}&-x_{2}&+x_{3}&&-b_{2}\\ +2x_{1}&+x_{2}&-x_{3}&-2x_{4}&&-b_{3}\\ +x_{1}&-x_{2}&+2x_{4}&&-b_{4}&=&0\\ +x_{1}&+x_{3}&-2x_{4}&&-b_{5}&&<=&0\\ +x_{2}&+x_{3}&+x_{4}&&-b_{6}&&<=&0\\ -x_{1}&-x_{2}&&+x_{4}&&-b_{7}&<=&0\\ +x_{1}&&-x_{3}&-2x_{4}&&-b_{8}&<=&0.\end{array}  \tag{54}\]  
What is behind this idea is to make all unknowns of the same type and not making differences among them. Once this is achieved, the resulting system is a homogeneous one, and we can solve it using the dual cone approach, obtaining the following general solution:  
\[ \begin{array}{r}\left(\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ b_{1}\\ b_{2}\\ b_{3}\\ b_{4}\\ b_{5}\\ b_{6}\\ b_{7}\\ b_{8}\end{array}\right)&=\left(\begin{array}{rrrrrr}2&1&-2&1\\ -1&-3&-4&-3\\ 0&5&0&-5\\ 1&-2&-1&-2\\ -2&-6&-3&4\\ 5&10&0&0\\ 5&0&0&0\\ 0&0&-5&-10\\ 0&0&5&0\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&5&10\\ 0&0&0&0\\ 0&0&0&0\\ \end{array}\right)_{\alpha}\end{array}  \tag{55}\]  
where the polytope part reduces to the zero vector, and, then, it can be removed from (55). We note that this process is reversible, that is, given (55), one can immediately obtain the system of linear inequalities (54) or an equivalent one to it. To this end, we can use the dual cone gamma algorithm, to obtain the dual of the cone resulting from (54) after adding one new component to the vectors in (54), which is zero in the first row for the column vectors in the linear space, \(\rho\), and acute cone, \(\pi\), matrices, and one for the column vectors in the polytope, \(\lambda\), parts, and considering the polytope resulting part as added to the resulting cone part. The general solution (55) integrates all the unknowns in the initial system with the unknowns of the independent \(b\) terms. This is a very interesting view because it does not differentiate the unknowns \(x\) and \(b\) apart from the fact that the \(x\) appear in several inequalities and the \(b\) in only one of them. The fact that the \(b\) coefficients are one is in principle unimportant, because we can divide the equations by any positive number, and then, we can normalize other unknowns. In order to be sure that we are dealing with a compatible system (53), we need to consider only the last eight unknowns in (55) and choose \(\rho,\pi\in\mathbf{R}\) and \(\pi\geq 0\) coefficients and apply these formulas for selecting \(b_{1},b_{2},\ldots,b_{8}\). Once these coefficients have been obtained, which guarantees compatibility, the set of all solutions of the system can immediately be obtained by using the first four inequalities. Note that we obtain the set of all solutions if they exist, and not just one solution. It is interesting to realize that \(b_{4},b_{5},b_{7}\), and \(b_{8}\) depend only on \(\rho_{1},\rho_{2},\rho_{3}\), and \(\rho_{4}\), respectively, because only one component of the corresponding rows in the \(\rho\) matrix is non-null, and all the components of the corresponding rows in the \(\pi\) and \(\lambda\) matrices are null. This means that the \(\rho_{1},\rho_{2},\rho_{3}\), and \(\rho_{4}\) of the linear space part can be immediately written in terms of \(b_{4},b_{5},b_{7}\), and \(b_{8}\). In addition, the \(\pi\) matrix has one column vector with all null components but a single one, which is \(1\) and is associated with the unknown \(b_{6}\). This means that \(\pi_{1}\) of the cone part of the solution can also be immediately written in terms of \(b_{6}\). In fact, we can select the values of any set of \(x\) unknowns or even a combination of \(x\) and \(b\) unknowns, and obtain the solutions for the remaining set of them. This can have important applications. For example, in a neural network built with these systems, we could choose as inputs and outputs any sets of neurons, that is, not necessarily those corresponding to the \(x\) as inputs and to the \(b\) as outputs. In other words, in neural networks, the input and output data need not be differentiated from the intermediate neuron values, as suggested if this approach is adopted.