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The Hirsch Conjecture stated that any d-dimensional polytope with n facetshas 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 pivotalgorithms 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 facetpivot method and perform some numerical tests to demonstrate its superiority tothe existing vertex pivot method. In the second part of this series, we showthat the proposed facet pivot method can solve the canonical linear programmingproblem in the worst case in at most n - d facet pivot iterations. This seriesof the papers was inspired by Smale's Problem 9 (Mathematical problems for thenext century, In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. Mathematics:frontiers and perspectives, American Mathematical Society, 271-294, 1999).

On the facet pivot simplex method for linear programming I: algorithms and numerical test