Editing Simplex algorithm (section)

==Overview==
{{further|Linear programming}}
[[Image:Simplex-description-en.svg|thumb|240px|A [[system of linear inequalities]] defines a [[polytope]] as a feasible region. The simplex algorithm begins at a starting [[vertex (geometry)|vertex]] and moves along the edges of the polytope until it reaches the vertex 
of the optimal solution.]]

[[Image:Simplex-method-3-dimensions.png|thumb|240px|Polyhedron of simplex algorithm in 3D]]

The simplex algorithm operates on linear programs in the [[canonical form]]

:maximize <math display="inline">\mathbf{c^T} \mathbf{x}</math>
:subject to <math>A\mathbf{x} \leq \mathbf{b}</math> and <math>\mathbf{x} \ge 0</math>

with <math>\mathbf{c} = (c_1,\, \dots,\, c_n)</math> the coefficients of the objective function, <math>(\cdot)^\mathrm{T}</math> is the [[matrix transpose]], and <math> \mathbf{x} = (x_1,\, \dots,\, x_n)</math> are the variables of the problem, <math>A</math> is a ''p''×''n'' matrix, and <math> \mathbf{b} = (b_1,\, \dots,\, b_p)</math>. There is a straightforward process to convert any linear program into one in standard form, so using this form of linear programs results in no loss of generality.

In geometric terms, the [[feasible region]] defined by all values of <math>\mathbf{x}</math> such that <math display="inline">A\mathbf{x} \le \mathbf{b}</math> and <math>\forall i, x_i \ge 0 </math> is a (possibly unbounded) [[convex polytope]]. An extreme point or vertex of this polytope is known as ''[[basic feasible solution]]'' (BFS).

It can be shown that for a linear program in standard form, if the objective function has a maximum value on the feasible region, then it has this value on (at least) one of the extreme points.<ref>{{harvtxt|Murty|1983|loc=Theorem 3.3}}</ref> This in itself reduces the problem to a finite computation since there is a finite number of extreme points, but the number of extreme points is unmanageably large for all but the smallest linear programs.<ref>{{harvtxt|Murty|1983|loc=Section 3.13|p=143}}</ref>
 
It can also be shown that, if an extreme point is not a maximum point of the objective function, then there is an edge containing the point so that the value of the objective function is strictly increasing on the edge moving away from the point.<ref name="Murty137">{{harvtxt|Murty|1983|loc=Section 3.8|p=137}}</ref> If the edge is finite, then the edge connects to another extreme point where the objective function has a greater value, otherwise the objective function is unbounded above on the edge and the linear program has no solution. The simplex algorithm applies this insight by walking along edges of the polytope to extreme points with greater and greater objective values. This continues until the maximum value is reached, or an unbounded edge is visited (concluding that the problem has no solution). The algorithm always terminates because the number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of the objective function), we hope that the number of vertices visited will be small.<ref name="Murty137"/>

The solution of a linear program is accomplished in two steps. In the first step, known as Phase I, a starting extreme point is found. Depending on the nature of the program this may be trivial, but in general it can be solved by applying the simplex algorithm to a modified version of the original program. The possible results of Phase I are either that a basic feasible solution is found or that the feasible region is empty. In the latter case the linear program is called ''infeasible''. In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point. The possible results from Phase II are either an optimum basic feasible solution or an infinite edge on which the objective function is unbounded above.<ref name="DantzigThapa1">[[George B. Dantzig]] and Mukund N. Thapa. 1997. ''Linear programming 1: Introduction''. Springer-Verlag.</ref><ref name="NeringTucker"/><ref name="Vanderbei">Robert J. Vanderbei, [http://www.princeton.edu/~rvdb/LPbook/ ''Linear Programming: Foundations and Extensions''], 3rd ed., International Series in Operations Research & Management Science, Vol. 114, Springer Verlag, 2008. {{isbn|978-0-387-74387-5}}. <!-- (An on-line second edition was formerly available. Vanderbei's site still contains extensive materials.) --></ref>