Is it possible that entering and leaving variables are same in simplex method?

Question

In simplex method, is it possible that entering and leaving variables are same? When can this happen?. If it cannot happen, what stops it from happening.

Answer 1

Yes it can happen. Two different ways.

(1) The gist of this way is quite simple. As the simplex progresses it ultimately makes smaller and smaller adjustments as it converges to a solution. Ultimately the change can be so small that is is smaller than the precision of the value(s) will allow. So the small change is a rounding error problem.

What should stop this is a convergence test to be sure that the adjustments are still significant.

(2) The n-dimensional simplex can be "hung" on the n+1 dimensional surface where the n+1 different points all yield essentially the same deviation. The way to avoid the problem is to start the simplex with a different set of initial conditions and see if it converges to the same solution.