The row pivoting method for linear programming

Solving linear programming is essentially to solve a special system of linear inequalities. Unlike other pivoting methods, we find that row geometry can fully and effectively exploit huge potential of Farkas Lemma in solving systems of linear inequalities and is a feasible way to solve linear programming. Therefore, we develop the row pivoting method for solving linear programming. The central idea of this method is to solve a system of linear inequalities corresponding to constraints of linear programming while keeping the optimality condition true all the time. In the proposed method, any linear programming problem can be solved without imposing redundancy or consistency assumptions, equations and inequalities in the constraints can be directly expressed in row vector form free of any auxiliary variables. The method can identify inconsistency and redundancy of constraints inherently, start with an arbitrary basic solution directly, eliminate equation constraints efficiently, and treat lower and upper bounds on any inequality constraint simultaneously. The proof of existence and convergence guarantees that the method can determine whether there exists an optimal solution to linear programming in finitely many steps.