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.^[1]

Problem formulation

For the rest of the discussion, it is assumed that a linear programming problem has been converted into the following standard form:

\begin{array}{rl} \text{minimize} & \boldsymbol{c}^{\mathrm{T}} \boldsymbol{x} \\ \text{subject to} & \boldsymbol{Ax} = \boldsymbol{b}, \boldsymbol{x} \ge \boldsymbol{0} \end{array}

where $A \in R m \times n$ . 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 $x \geq 0$ 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.

Algorithmic description

Optimality conditions

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

\begin{align} \boldsymbol{Ax} & = \boldsymbol{b}, \\ \boldsymbol{A}^{\mathrm{T}} \boldsymbol{\lambda} + \boldsymbol{s} & = \boldsymbol{c}, \\ \boldsymbol{x} & \ge \boldsymbol{0}, \\ \boldsymbol{s} & \ge \boldsymbol{0}, \\ \boldsymbol{s}^{\mathrm{T}} \boldsymbol{x} & = 0 \end{align}

where $λ$ and $s$ are the Lagrange multipliers associated with the constraints $Ax = b$ and $x \geq 0$ , respectively.^[2] The last condition, which is equivalent to $s i x i = 0$ 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.^{[lower-alpha 1]} Since $A$ has full rank, $B$ is nonsingular. Without loss of generality, assume that $A = [B N]$ . Then $x$ is given by

\boldsymbol{x} = \begin{bmatrix} \boldsymbol{x_B} \\ \boldsymbol{x_N} \end{bmatrix} = \begin{bmatrix} \boldsymbol{B}^{-1} \boldsymbol{b} \\ \boldsymbol{0} \end{bmatrix}

where $x B \geq 0$ . Partition $c$ and $s$ accordingly into

\begin{align} \boldsymbol{c} & = \begin{bmatrix} \boldsymbol{c_B} \\ \boldsymbol{c_N} \end{bmatrix}, \\ \boldsymbol{s} & = \begin{bmatrix} \boldsymbol{s_B} \\ \boldsymbol{s_N} \end{bmatrix}. \end{align}

To satisfy the complementary slackness condition, let $s B = 0$ . It follows that

\begin{align} \boldsymbol{B}^{\mathrm{T}} \boldsymbol{\lambda} & = \boldsymbol{c_B}, \\ \boldsymbol{N}^{\mathrm{T}} \boldsymbol{\lambda} + \boldsymbol{s_N} & = \boldsymbol{c_N}, \end{align}

which implies that

{\begin{aligned}{\boldsymbol {\lambda }}&=({\boldsymbol {B}}^{{{\mathrm {T}}}})^{{-1}}{\boldsymbol {c_{B}}},\\{\boldsymbol {s_{N}}}&={\boldsymbol {c_{N}}}-{\boldsymbol {N}}^{{{\mathrm {T}}}}{\boldsymbol {\lambda }}.\end{aligned}}

If $s N \geq 0$ at this point, the KKT conditions are satisfied, and thus $x$ is optimal.

Pivot operation

If the KKT conditions are violated, a pivot operation consisting of introducing a column of $N$ into the basis at the expense of an existing column in $B$ is performed. In the absence of degeneracy, a pivot operation always results in a strict decrease in $c T x$ . 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.^[4]

Select an index $m < q \leq n$ such that $s q < 0$ as the entering index. The corresponding column of $A$ , $A q$ , will be moved into the basis, and $x q$ will be allowed to increase from zero. It can be shown that

\frac{\partial (\boldsymbol{c}^{\mathrm{T}} \boldsymbol{x})}{\partial x_q} = s_q,

i.e., every unit increase in $x q$ will results in a decrease by $- s q$ in $c T x$ .^[5] Since

\boldsymbol{B x_B} + \boldsymbol{A}_q x_q = \boldsymbol{b},

$x B$ must be correspondingly decreased by $Δ x B = B -1 A q x q$ subject to $x B - Δ x B \geq 0$ . Let $d = B -1 A q$ . If $d \leq 0$ , no matter how much $x q$ is increased, $x B - Δ x B$ will stay nonnegative. Hence, $c T x$ can be arbitrarily decreased, and thus the problem is unbounded. Otherwise, select an index $p = argmin 1\leq i \leq m {x i / d i | d i > 0}$ as the leaving index. This choice effectively increases $x q$ from zero until $x p$ is reduced to zero while maintaining feasibility. The pivot operation concludes with replacing $A p$ with $A q$ in the basis.

Numerical example

Practical issues

Degeneracy

Because the revised simplex method is mathematically equivalent to the simplex method, it also suffers from degeneracy, where a pivot operation does not result in a decrease in $c T x$ , and a chain of pivot operations causes the basis to cycle. A perturbation or lexicographic strategy can be used to prevent cycling and guarantee termination.^[6]

Basis representation

Two types of linear systems involving $B$ are present in the revised simplex method:

\begin{align} \boldsymbol{B z} & = \boldsymbol{y}, \\ \boldsymbol{B}^{\mathrm{T}} \boldsymbol{z} & = \boldsymbol{y}. \end{align}

Instead of refactorizing $B$ , usually an LU factorization is directly updated after each pivot operation, for which purpose there exist several strategies such as the Forrest−Tomlin and Bartels−Golub methods. However, the amount of data representing the updates as well as numerical errors builds up over time and makes periodic refactorization necessary.^[1]^[7]

Notes and references

Notes

↑ The same theorem also states that the feasible polytope has at least one vertex and that there is at least one vertex which is optimal.^[3]

References

1 2 Morgan 1997, §2.
↑ Nocedal & Wright 2006, p. 358, Eq. 13.4.
↑ Nocedal & Wright 2006, p. 363, Theorem 13.2.
↑ Nocedal & Wright 2006, p. 370, Theorem 13.4.
↑ Nocedal & Wright 2006, p. 369, Eq. 13.24.
↑ Nocedal & Wright 2006, p. 381, §13.5.
↑ Nocedal & Wright 2006, p. 372, §13.4.

Bibliography

Morgan, S. S. (1997). A Comparison of Simplex Method Algorithms (MSc thesis). University of Florida. Archived from the original on 7 August 2011.
Nocedal, J.; Wright, S. J. (2006). Mikosch, T. V.; Resnick, S. I.; Robinson, S. M., eds. Numerical Optimization. Springer Series in Operations Research and Financial Engineering (2nd ed.). New York, NY, USA: Springer. ISBN 978-0-387-30303-1.

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