Linear Programs and Their Solution Sets
The polyhedral cone feasibility problem PCFP that occupied us in the last two chapters, though fundamental, is better understood when regarded within the more general context of linear programming. Succinctly described, the latter is a family of problems that consist in optimizing (i.e., maximizing or minimizing) a linear function over a set defined by linear constraints (equalities and/or inequalities).
A first step towards the solution of such a problem requires one to decide whether the family of constraints is satisfiable, that is, whether it defines a nonempty set. The polyhedral cone feasibility problem is a particular case of such a requirement.
Interestingly, optimization and feasibility problems appear to reduce to one another. Thus, in this chapter we solve PCFP by recasting it as an optimization problem. Conversely, in a subsequent chapter, we will reduce the solution of optimization problems to a sequence of instances of PCFP.
Because of these considerations, before proceeding with the exposition of new algorithms, we make a pause and devote it to the understanding of linear programs and their sets of solutions. As usual, such an understanding proves of the essence at the moment of defining condition.