Simplex algorithm

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In mathematical optimization theory, the simplex algorithm, created by the American mathematician George Dantzig in 1947, is a popular algorithm for numerical solution of the linear programming problem. The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the century.[1]

An unrelated, but similarly named method is the Nelder-Mead method or downhill simplex method due to Nelder & Mead (1965) and is a numerical method for optimising many-dimensional unconstrained problems, belonging to the more general class of search algorithms.

In both cases, the method uses the concept of a simplex, which is a polytope of N + 1 vertices in N dimensions: a line segment in one dimension, a triangle in two dimensions, a tetrahedron in three-dimensional space and so forth.

Contents

[edit] Description

Main article: Linear programming
A series of linear inequalities defines a polytope as a feasible region. The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimum solution.
A series of linear inequalities defines a polytope as a feasible region. The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimum solution.

A linear programming problem consists of a collection of linear inequalities on a number of real variables and a given linear function (on these real variables) which is to be maximized or minimized. Further details are given in the linear programming article.

In geometric terms we are considering a closed convex polytope, P, defined by intersecting a number of half-spaces in n-dimensional Euclidean space; each half-space is the area which lies on one side of a hyperplane. If the objective is to maximise a linear functional L(x), consider the hyperplanes H(c) defined by L(x) = c; as c increases, these form a parallel family. If the problem is well-posed, we want to find the largest value of c such that H(c) intersects P (if there is no such largest value of c, this isn't a reasonable question for optimization as it stands). In this case we can show that the optimum value of c is attained on the boundary of P. Methods for finding this optimum point on P work in several ways: some attempt to improve a possible point by moving through the interior of P (so-called interior point methods); others start and remain on the boundary searching for an optimum.

The simplex algorithm follows this latter methodology. The idea is to move along the facets of P in search of the optimum, from point to point. Note that, unless the optimum occurs on an edge or face that is parallel to H, the optimum will be unique and occur at a vertex of the polytope. If an optimum is found on an edge or face that is parallel to H then the optimum is not unique and can be obtained at any point on the edge or face. Since the simplex algorithm is concerned only with finding a single optimal point (even if other equally-optimal points exist), it is possible to look solely at moves skirting the edge of a simplex, ignoring the interior. The algorithm specifies how this is to be done.

We start at some vertex of the polytope, and at every iteration we choose an adjacent vertex such that the value of the objective function does not decrease. If no such vertex exists, we have found a solution to the problem. But usually, such an adjacent vertex is nonunique, and a pivot rule must be specified to determine which vertex to pick. Various pivot rules exist.

In 1972, Klee and Minty [2] gave an example of a linear programming problem in which the polytope P is a distortion of an n-dimensional cube. They showed that the simplex method as formulated by Dantzig visits all 2n vertices before arriving at the optimal vertex. This shows that the worst-case complexity of the algorithm is exponential time. Similar examples have been found for other pivot rules. It is an open question if there is a pivot rule with polynomial time worst-case complexity.

Nevertheless, the simplex method is remarkably efficient in practice. Attempts to explain this employ the notion of average complexity or (recently) smoothed complexity.

Other algorithms for solving linear programming problems are described in the linear programming article.

[edit] Problem input

Consider a linear programming problem,

maximize \mathbf{c}^T \mathbf{x}
subject to \mathbf{A}\mathbf{x} \le \mathbf{b}, \, \mathbf{x} \ge 0.

The simplex algorithm requires the linear programming problem to be in augmented form. The problem can then be written as follows in matrix form:

Maximize Z in:
\begin{bmatrix} 1 & -\mathbf{c}^T & 0 \\ 0 & \mathbf{A} & \mathbf{I} \end{bmatrix} \begin{bmatrix} Z \\ \mathbf{x} \\ \mathbf{x}_s \end{bmatrix} = \begin{bmatrix} 0 \\ \mathbf{b} \end{bmatrix}
\mathbf{x}, \, \mathbf{x}_s \ge 0

where x are the variables from the standard form, xs are the introduced slack variables from the augmentation process, c contains the optimization coefficients, A and b describe the system of constraint equations, and Z is the variable to be maximized.

The system is typically underdetermined, since the number of variables exceed the number of equations. The difference between the number of variables and the number of equations gives us the degrees of freedom associated with the problem. Any solution, optimal or not, will therefore include a number of variables of arbitrary value. The simplex algorithm uses zero as this arbitrary value, and the number of variables with value zero equals the degrees of freedom.

Variables of nonzero value are called basic variables, and variables of zero value are called nonbasic variables in the simplex algorithm. [This definition is problematic, since basic variables can also take zero value.]

This form simplifies finding the initial basic feasible solution (BF), since all variables from the standard form can be chosen to be nonbasic (zero), while all new variables introduced in the augmented form are basic (nonzero), since their value can be trivially calculated (\mathbf{x}_{s\,i} = \mathbf{b}_{i} for them, since the augmented problem matrix is diagonal on its right half).

[edit] Revised simplex algorithm

[edit] Matrix form of the simplex algorithm

At any iteration of the simplex algorithm, the tableau will be of this form:

\begin{bmatrix} 1 & \mathbf{c}_B^T \mathbf{B}^{-1}\mathbf{A} -\mathbf{c}^T & \mathbf{c}_B^T \mathbf{B}^{-1} \\ 0 & \mathbf{B}^{-1}\mathbf{A} & \mathbf{B}^{-1} \end{bmatrix} \begin{bmatrix} Z \\ \mathbf{x} \\ \mathbf{x}_s \end{bmatrix} = \begin{bmatrix} \mathbf{c}_B^T \mathbf{B}^{-1} \mathbf{b} \\ \mathbf{B}^{-1}\mathbf{b} \end{bmatrix}

where \mathbf{c}_B are the coefficients of basic variables in the c-matrix; and B is the columns of [\mathbf{A} \, \mathbf{I}] corresponding to the basic variables.

It is worth noting that B and \mathbf{c}_B are the only variables in this equation (except Z and x of course). Everything else is constant throughout the algorithm.

[edit] Algorithm

  • Choose a basic feasible solution (BF) as shown above
This implies that B is the identity matrix, and \mathbf{c}_B is a zero-vector.
  • While nonoptimal solution:
    • Determine direction of highest gradient
    Choose the variable associated with the coefficient in \mathbf{c}_B^{T} \mathbf{B}^{-1}\mathbf{A} -\mathbf{c}^{T} that has the highest negative magnitude. This nonbasic variable, which we call the entering basic, will be increased to help maximize the problem.
    • Determine maximum step length
    Use the \begin{bmatrix} \mathbf{B}^{-1}\mathbf{A} & \mathbf{B}^{-1} \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ \mathbf{x}_s \end{bmatrix} = \mathbf{B}^{-1}\mathbf{b} subequation to determine which basic variable reaches zero first when the entering basic is increased. This variable, which we call the leaving basic then becomes nonbasic. This operation only involves a single division for each basic variable, since the existing basic-variables occur diagonally in the tableau.
    • Rewrite problem
    Modify B and \mathbf{c}_B to account for the new basic variables. This will automatically make the tableau diagonal for the existing and new basic variables.
    • Check for improvement
    Repeat procedure until no further improvement is possible, meaning all the coefficients of \mathbf{c}_B^{T} \mathbf{B}^{-1}\mathbf{A} -\mathbf{c}^{T} are positive. Procedure is also terminated if all coefficients are zero, and the algorithm has walked in a circle and revisited a previous state.

[edit] Common Implementations of The Revised Simplex Method

  • The Bartels-Golub Method
  • The Sparse Bartels-Golub Method
  • The Forrest-Tomlin Method
  • Reid’s Method

[edit] References

  • Greenberg, Harvey J., Klee-Minty Polytope Shows Exponential Time Complexity of Simplex Method University of Colorado at Denver (1997) PDF download
  • Frederick S. Hillier and Gerald J. Lieberman: Introduction to Operations Research, 8th edition. McGraw-Hill. ISBN 0-07-123828-X
  • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 29.3: The simplex algorithm, pp.790–804.
  • Hamdy A. Taha: Operations Research: An Introduction, 8th ed., Prentice Hall, 2007. ISBN 0-13-188923-0
  • Richard B. Darst: Introduction to Linear Programming: Applications and Extensions

[edit] Note

  1. ^ Computing in Science and Engineering, volume 2, no. 1, 2000
  2. ^ Greenberg, cites: V. Klee and G.J. Minty. "How Good is the Simplex Algorithm?" In O. Shisha, editor, Inequalities, III, pages 159–175. Academic Press, New York, NY, 1972

[edit] See also

Wikibooks

[edit] External links