among all nonzero $n \times n$ sub-determinants of $\A$.

Finding the diameter of convex polytopes provides only a surmised lowest iteration number for which an optimal pivot rule may achieve. Finding actually such a pivot rule (the way to choose the next neighbor vertex) is also a difficult problem. Researchers proposed many pivot rules with the hope that they may achieve the number of iterations (in the worst case) bounded by a polynomial (see [@tz93] and references therein). However, since Klee and Minty [@km72] constructed a cube and showed that Dantzig’s rule needs an exponential number of iterations in the worst case to solve the Klee and Minty cube problem, people have showed similar results for almost every popular pivot rule [@friedmann11; @gs79; @jeroslow73; @psz08]. It is now believed that finding a pivot rule that will solve all linear programming problems in the worst case in polynomial time is a very difficult problem [@smale99].

Existing pivot rules consider one of many merit criteria to select an entering variable, such as the most negative index in the reduced cost vector (Dantzig’s rule), the best improvement rule, Bland’s least index pivoting rule, the steepest edge simplex rule, Zadeh’s rule, among others [@tz93]. 