How to pivot to an adjacent vertex in simplex method

Question

In the simplex method, we need to move from one vertex of the polyhedron to an adjacent one. Suppose the polyhedron is $P=\{x\in\mathbb{R}^n\mid Ax=b,x\geq0\}$ with rank$A=m<n$. For a nondegenerate vertex, it has $n$ adjacent vertices. In the simplex textbooks, they say that to get an adjacent vertex, we just need to switch one basic variable with one nonbasic variable. However, there are $m$ basic variables and $n−m$ nonbasic variables. Therefore, the total number of such switches is $m(n−m)$. But I know, there are only $n$ adjacent vertices. So am I missing something?

Answer 1

Since every vertex of $P$ is basic feasible solution and vice versa, a non-degenerate vertex has exactly $m(n-m)$ adjacent vertices. I don't know what gave you the idea that this number is $n$, but this is simply not the case.