On purpose, I do not write details about it, because I am interested in the intuition and not the details. But if someone asks for it, I can do that. So my question: Could anyone help me with the lexicographic pivoting rule for degenerate problems with simplex algorithm.
So the problem is when we are at a degenerate basic feasible solution $x^*$. This means that for a basic variable $ i \in B$ we have that $x_i^* = 0$. This means that the step length $\lambda$ will always be $0$, in order to remain primal feasible (when we want to go to $x^* + \lambda d$ for the $j$-th non-basic direction.
So how can we make progress with choosing another pivot? Anyhow, the step length will be $0$ again, since $$\lambda:= \min \left \{ \frac{x_i^*}{ (A_B^{-1}A)_{ij}} \colon (A_B^{-1}A)_{ij} > 0 \right \},$$
Since $x_i^*=0$ for some $i \in B$.
Thank you very much for your help.
I will also avoid details. If you are using classical simplex method, then it can easily happen that you will come back to the same simplex table. (Actually it happens very often in the case of transportation problems, when the amounts are small integers.) So you would like to avoid this. You can avoid it in such manner that you introduce some ordering on the variables. After that you will track some vector, and in each step you will choose the pivot in such a manner that your vector is lexicographically increasing. This prevents the situation, that you will come back into the same state, like before.
If you would like to interpret it geometrically, then degeneracy means, that you have more supporting hyperplanes through a concrete point, than it is necessary. In one step of the simplex method you are just changing the set of supporting hyperplanes by one, but you are staying in the same point. Therefore, if you would not like to come back to the same set of supporting hyperplanes, then you need an ordering on the hyperplanes. And exactly this is implemented in the lexicographic method.