Solving Systems of Equations

Abstract

The information about linear programming problems developed up to this point has revealed the need to know an efficient method for solving systems of linear equations. In fact, the main results achieved in previous chapters can be summarized as follows: 1. Optimal solutions occur always at an extreme point of the primal feasible region, as illustrated in the output space diagram of figure 5.1. 2. To each extreme point in the output space there corresponds a feasible basis (a cone that contains the RHS vector b) in the input space, as in figure 5.2. 3. To solve a linear programming problem, therefore, it is sufficient to explore only extreme points that, in an output space diagram similar to figure 5.1, can be identified only in qualitative terms. 4. An extreme point of the primal feasible solution region is associated with a basic system of equations (see section 5.4) whose solution provides a basic feasible solution in quantitative terms. In the intuitive jargon developed previously, a basic system of linear equations is composed of three parts: an object to be measured, a ruler for measuring the object, and the measurement of the object. 5. As a consequence, we will discuss a solution procedure for linear programming problems called the simplex method, which is organized around a process of solving a sequence of systems of linear equations. A member of this sequence is called an iteration.