# 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
The interesting case of solving simultaneous subsystems of given systems and the corresponding duals is dealt with in [20]. Example of a Complete System of Linear Inequalities  
To solve the complete system of linear inequalities:  
\[ \begin{array}{rrrrr}x_{1}&+x_{2}&-x_{4}&\leq&3\\ -2x_{1}&-2x_{2}&-x_{3}&+x_{4}&\leq&-3\\ -2x_{1}&-x_{2}&-x_{3}&+x_{4}&\leq&-2\\ -2x_{1}&-x_{2}&-x_{3}&+x_{4}&\leq&-2\\ -x_{3}&-x_{4}&\leq&-1\\ \end{array}  \tag{29}\]  
we incorporate the fictitious unknown \(x_{5}\), the constraint \(x_{5}=1\), to avoid altering the system, and the redundant constraint \(-x_{5}\leq 0\), as the first inequality, and obtain:  
\[ \begin{array}{rrrrr}&&-x_{5}&\leq&0\\ x_{1}&+x_{2}&-x_{4}&-3x_{5}&\leq&0\\ -2x_{1}&-2x_{2}&-x_{3}&+x_{4}&+3x_{5}&\leq&0\\ -2x_{1}&-x_{2}&-x_{3}&+x_{4}&+2x_{5}&\leq&0\\ &-x_{3}&-x_{4}&+x_{5}&\leq&0\\ -x_{1}&+x_{2}&-2x_{3}&-x_{4}&+x_{5}&\leq&0\\ &&x_{5}&=&1.\\ \end{array}  \tag{30}\]  
We solve first the homogeneous system and impose later the last constraint. The solution of the homogeneous system is the dual cone of the cone generated by the vectors with components given by the inequality coefficients, that is, the dual of the cone:  
\[\mathbf{A}_{\pi}=\left( \begin{array}{rrrrrrr}0&1&-2&-2&0&-1\\ 0&1&-2&-1&0&1\\ 0&0&-1&-1&-1&-2\\ 0&-1&1&1&-1&-1\\ -1&-3&3&2&1&1\\ \end{array} \right)_{\pi}\, \tag{31}\]  
where the coefficients are the columns, which is the cone:  
\[\mathbf{A}_{\pi}^{p}=\left( \begin{array}{rrrrr}1\\ 0\\ -1\\ 1\\ 0\end{array} \right)_{\rho}+\left( \begin{array}{rrrrr}-1&1&-1&0&1\\ 0&-1&1&1&2\\ 2&0&1&1&1\\ 0&0&0&0&0&0\\ 0&0&0&1&1\\ \end{array} \right)_{\pi}\, \tag{32}\]  
which has been created using the \(\Gamma\) algorithm described in [6; 18]. If we force now the constraint \(x_{5}=1\), we obtain \(\pi_{4}+\pi_{5}=1\), that is, the coefficients \(\pi_{4}\) and \(\pi_{5}\) add up to one, and then, they become lambda coefficients \(\lambda_{1}\) and \(\lambda_{2}\) because of its unit sum. Then, the solution of the initial system, after removing the fictitious unknown \(x_{5}\), is:  
\[\left( \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{array} \right)=\left( \begin{array}{c}1\\ 0\\ -1\\ 1\end{array} \right)_{\rho}+\left( \begin{array}{ccc}-1&1&1\\ 0&-1&1\\ 2&0&1\\ 0&0&0\end{array} \right)_{\pi}+\left( \begin{array}{ccc}0&1\\ 1&2\\ 1&1\\ 0&0\end{array} \right)_{\lambda}=\left( \begin{array}{c}\rho-\pi_{1}+\pi_{2}- \pi_{3}+\lambda_{2}\\ -\pi_{2}+\pi_{3}+\lambda_{1}+\lambda_{2}\\ -\rho+2\pi_{2}+\pi_{3}+\lambda_{1}+\lambda_{2}\\ \rho\end{array} \right) \tag{33}\]  
where the last term refers to the polytope generated by the vertices defined as its column vectors, and \(\lambda\) refers to linear convex combinations, that is, to non-negative real numbers adding to one. To check, we can replace these values in the initial system of equations (29) to obtain:  
\[\begin{array}{  
Thus, to see if this particular case of a complete system is compatible in this case, it is sufficient to check whether or not the vector of the independent terms belongs to the cone generated by the column vectors of the inequality system, for which it is necessary to calculate the dual cone generated by these column vectors and see if the vector of the independent terms belongs to it. To study the compatibility of a complete system of any general type, that is, with systems of inequalities \"less than or equal to\", \"greater than or equal to\", and \"equal to\",and with unrestricted variables in sign, it is sufficient to rewrite the initial system as another equivalent one that meets the conditions of the particular case, proceeding as follows:  
* Replace each inequality \"greater than or equal\" with an \"equality\" by simply subtracting a new non-negative variable, one per each inequality. * Replace each inequality \"less than or equal\" with an \"equality\", a new non-negative variable, one per inequality. * Replace all non-positive unknowns with others equal to themselves but with the sign changed.