Revised simplex method

The matrix-oriented approach allows for greater computational efficiency by enabling sparse matrix operations.

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.

Consider a linear program where Let initially, which corresponds to a feasible vertex x = [0 0 0 10 15]T. At this moment, Choose q = 3 as the entering index.

However, the amount of data representing the updates as well as numerical errors builds up over time and makes periodic refactorization necessary.