In the previous part we implemented and tested the simplex method on a simple example, and it has executed without any problems. In the first part, we have seen an example of the unbounded linear program.
What will happen if we apply the simplex algorithm for it?
It may even happen that some tableau is repeated in a sequence of degenerate pivot steps.
It may even happen that some tableau is repeated in a sequence of degenerate pivot steps, and so the algorithm might pass through an infinite sequence of tableau without any progress. A pivot rule is a rule for selecting the entering variable if there are several possibilities, which is usually the case(in our algorithm determine this element).
If the edge is finite, then the edge connects to another extreme point where the objective function has a greater value, otherwise the objective function is unbounded above on the edge and the linear program has no solution.
The simplex algorithm applies this insight by walking along edges of the polytope to extreme points with greater and greater objective values.Without an objective, a vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself.Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.There are several ways that probably avoid cycling.One of them is the already mentioned Bland’s rule, but it is one of the slowest pivot rules and it is almost never used in practice.This in itself reduces the problem to a finite computation since there is a finite number of extreme points, but the number of extreme points is unmanageably large for all but the smallest linear programs.It can also be shown that, if an extreme point is not a maximum point of the objective function, then there is an edge containing the point so that the objective function is strictly increasing on the edge moving away from the point.There is a straightforward process to convert any linear program into one in standard form, so using this form of linear programs results in no loss of generality.In geometric terms, the feasible region defined by all values of is a (possibly unbounded) convex polytope.On this example, we can see that on first iteration objective function value made no gains.In general, there might be longer runs of degenerate pivot steps.