  The Hirsch Conjecture stated that any d-dimensional polytope with n facets
has a diameter at most equal to n - d. This conjecture was disproven by Santos
(A counterexample to the Hirsch Conjecture, Annals of Mathematics, 172(1)
383-412, 2012). The implication of Santos' work is that all vertex pivot
algorithms cannot solve the linear programming problem in the worst case in n -
d vertex pivot iterations.
  In this paper, the first part in this series of papers, we propose a facet
pivot method and perform some numerical tests to demonstrate its superiority to
the existing vertex pivot method. In the second part of this series, we show
that the proposed facet pivot method can solve the canonical linear programming
problem in the worst case in at most n - d facet pivot iterations. This series
of the papers was inspired by Smale's Problem 9 (Mathematical problems for the
next century, In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. Mathematics:
frontiers and perspectives, American Mathematical Society, 271-294, 1999).
