Why do active set methods or the simplex method pivot only one variable at a time? Ostensibly, we could add multiple columns to the basis during pivoting, but the standard presentation of the methods don't do this. Likely, something breaks, but I don't understand what.
If something more concrete would help, on page 472 of Numerical Optimization, we have the active set method for a convex QP in Algorithm 16.3.
Pivoting occurs in the choice of $j$ and and that single index $j$ entering the working set $W$. Why choose a single $j$? Couldn't we choose multiple indices that correspond to multiple negative multipliers $\lambda$?