Revised simplex method
Without loss of generality, it is assumed that the constraint matrix A has full row rank and that the problem is feasible, i.e., there is at least one x ≥ 0 such that Ax = b.The KKT conditions of a linear programming problem in the standard form is where λ and s are the Lagrange multipliers associated with the constraints Ax = b and x ≥ 0, respectively.By what is sometimes known as the fundamental theorem of linear programming, a vertex x of the feasible polytope can be identified by being a basis B of A chosen from the latter's columns.Therefore, if the problem is bounded, the revised simplex method must terminate at an optimal vertex after repeated pivot operations because there are only a finite number of vertices.However, the amount of data representing the updates as well as numerical errors builds up over time and makes periodic refactorization necessary.
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