Linear systems of inequalities

Homogeneous linear systems of inequalities

In this leasson we explain how to solve an homogeneous system of inequalities, showing that the solution is a cone, whoich is the dual of the cone generated by the coefficients of the linear equations.

Solving complete linear systems of inequalities

In this lesson we explain how to solve a linear system of inequalities using the Gamma algorithm. It is shown that the general solution of these systems is the sum of a linear space plus a cone plus a polytope. One example of application is described in detail.

Compatibility of complete linear systems of inequalities

This lesson delas with the compatibility of a system of linear inequalities. These systems can have no solution, a unique solution or infinitely many solutions. An algorithm is presented that allows to know if the system has at least one solution, even in the case of the independent terms to be symbolic.

Equations of a polyhedron

In this lesson it is explained how the equations of a polyhedron are obtained. Since we have an algorithm to solve linear systems of inequalities and we know that its solution is a polyhedron, that is, a linea space plus a cone plus a polytope, we use an inverse algorithm to obtain a linea system of inequalities having the given polyhedron as its solution. An example of application is given.

Sets of solutions of linear systems of inequalities

This lesson describes the alhgebraic structure of the set of all solutions of linear systems of equations and of inequations. It is very important to know these structures because it helps to understand linear suystem more deeply. The relevance of this lesson adquires the highest degree when dealing with applications, where we need to identify the boundedness of unboundedness of the solutions and how the feasible set changes with new constraints.

Cone associated with a polytope. Facets of cones and polytopes

This lesson explains how to associate a cone to a given polytope. Since the Gamma algotithm provides the facets of all dimensions of a cone, this allows to obtain all facets of any dimension of any polytope.