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Revised simplex method
In mathematical optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming.
The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constraints adjusted to a set of basic variables, it maintains a representation of a basis of the matrix representing the constraints. The matrix-oriented approach allows for greater computational efficiency by enabling sparse matrix operations.
For the rest of the discussion, it is assumed that a linear programming problem has been converted into the following standard form:
where A ∈ ℝm×n. 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. If A is rank-deficient, either there are redundant constraints, or the problem is infeasible. Both situations can be handled by a presolve step.
For linear programming, the Karush–Kuhn–Tucker conditions are both necessary and sufficient for optimality. 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. The last condition, which is equivalent to sixi = 0 for all 1 < i < n, is called the complementary slackness condition.
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. Since A has full rank, B is nonsingular. Without loss of generality, assume that A = [B N]. Then x is given by
where xB ≥ 0. Partition c and s accordingly into
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Revised simplex method
In mathematical optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming.
The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constraints adjusted to a set of basic variables, it maintains a representation of a basis of the matrix representing the constraints. The matrix-oriented approach allows for greater computational efficiency by enabling sparse matrix operations.
For the rest of the discussion, it is assumed that a linear programming problem has been converted into the following standard form:
where A ∈ ℝm×n. 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. If A is rank-deficient, either there are redundant constraints, or the problem is infeasible. Both situations can be handled by a presolve step.
For linear programming, the Karush–Kuhn–Tucker conditions are both necessary and sufficient for optimality. 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. The last condition, which is equivalent to sixi = 0 for all 1 < i < n, is called the complementary slackness condition.
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. Since A has full rank, B is nonsingular. Without loss of generality, assume that A = [B N]. Then x is given by
where xB ≥ 0. Partition c and s accordingly into