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{{about|the linear programming algorithm|the non-linear optimization heuristic|Nelder–Mead method}}
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In [[optimization (mathematics)|mathematical optimization]], [[George Dantzig|Dantzig]]'s '''simplex algorithm''' (or '''simplex method''') is a popular [[algorithm]] for [[linear programming]].<ref name="Murty">{{cite book |last=Murty |first=Katta G. |author-link=Katta G. Murty |year=2000 |title=Linear programming |publisher=John Wiley & Sons
The name of the algorithm is derived from the concept of a [[simplex]] and was suggested by [[Theodore Motzkin|T. S. Motzkin]].<ref name="Murty22" >{{harvtxt|Murty|1983|loc=Comment 2.2}}</ref> Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial ''[[cone (geometry)|cone]]s'', and these become proper simplices with an additional constraint.<ref name="Murty39">{{harvtxt|Murty|1983|loc=Note 3.9}}</ref><ref name="StoneTovey">{{cite journal|last1=Stone|first1=Richard E.|last2=Tovey|first2=Craig A.|title=The simplex and projective scaling algorithms as iteratively reweighted least squares methods|journal=SIAM Review|volume=33|year=1991|issue=2|pages=220–237
|mr=1124362|jstor=2031142|doi=10.1137/1033049}}</ref><ref>{{cite journal|last1=Stone|first1=Richard E.|last2=Tovey|first2=Craig A.|title=Erratum: The simplex and projective scaling algorithms as iteratively reweighted least squares methods|journal=SIAM Review|volume=33|year=1991|issue=3|pages=461|mr=1124362|doi=10.1137/1033100|jstor=2031443}}</ref><ref name="Strang">{{cite journal|last=Strang|first=Gilbert|author-link=Gilbert Strang|title=Karmarkar's algorithm and its place in applied mathematics|journal=[[The Mathematical Intelligencer]]|date=1 June 1987|issn=0343-6993|pages=4–10|volume=9|doi=10.1007/BF03025891|mr=883185|issue=2|s2cid=123541868}}</ref> The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a [[polytope]]. The shape of this polytope is defined by the [[System of linear inequalities|constraints]] applied to the objective function.
==
[[George Dantzig]] worked on planning methods for the US Army Air Force during World War II using a [[Mechanical_calculator#1900s_to_1970s|desk calculator]]. During 1946, his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of [[Wassily Leontief]], however, at that time he didn't include an objective as part of his formulation. Without an objective, a vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself. Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.<ref>{{Cite journal|url = https://apps.dtic.mil/sti/pdfs/ADA112060.pdf|archive-url = https://web.archive.org/web/20150520183722/http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA112060|url-status = live|archive-date = May 20, 2015|title = Reminiscences about the origins of linear programming|date = April 1982|journal = Operations Research Letters|doi = 10.1016/0167-6377(82)90043-8|volume = 1|issue = 2 |pages=43–48|last1 = Dantzig|first1 = George B.}}</ref> Development of the simplex method was evolutionary and happened over a period of about a year.<ref>{{Cite journal|url = http://www.phpsimplex.com/en/Dantzig_interview.htm|title = An Interview with George B. Dantzig: The Father of Linear Programming|last = Albers and Reid|date = 1986|journal = College Mathematics Journal|volume = 17|issue = 4|doi = 10.1080/07468342.1986.11972971|pages = 292–314}}</ref>
After Dantzig included an objective function as part of his formulation during mid-1947, the problem was mathematically more tractable. Dantzig realized that one of the unsolved problems that [[George Dantzig#
==
{{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
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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>
==Standard form==
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When this process is complete the feasible region will be in the form
:<math>\mathbf{A}\mathbf{x} = \mathbf{b},\, \forall
It is also useful to assume that the rank of <math>\mathbf{A}</math> is the number of rows. This results in no loss of generality since otherwise either the system <math>\mathbf{A}\mathbf{x} = \mathbf{b}</math> has redundant equations which can be dropped, or the system is inconsistent and the linear program has no solution.<ref>{{harvtxt|Murty|1983|p=173}}</ref>
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\begin{bmatrix}
1 & -\mathbf{c}^T & 0 \\
\mathbf{0} & \mathbf{A} & \mathbf{b}
\end{bmatrix}
</math>
The first row defines the objective function and the remaining rows specify the constraints. The zero in the first column represents the zero vector of the same dimension as the vector
Conversely, given a basic feasible solution, the columns corresponding to the nonzero variables can be expanded to a nonsingular matrix. If the corresponding tableau is multiplied by the inverse of this matrix then the result is a tableau in canonical form.<ref>{{harvtxt|Murty|1983|loc=section 3.12}}</ref>
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===Entering variable selection===
Since the entering variable will, in general, increase from 0 to a positive number, the value of the objective function will decrease if the derivative of the objective function with respect to this variable is negative. Equivalently, the value of the objective function is
If there is more than one column so that the entry in the objective row is positive then the choice of which one to add to the set of basic variables is somewhat arbitrary and several ''entering variable choice rules''<ref name="Murty66">{{harvtxt|Murty|1983|p=66}}</ref> such as [[Devex algorithm]]<ref>Harris, Paula MJ. "Pivot selection methods of the Devex LP code." Mathematical programming 5.1 (1973): 1–28</ref> have been developed.
If all the entries in the objective row are less than or equal to 0 then no choice of entering variable can be made and the solution is in fact optimal. It is easily seen to be optimal since the objective row now corresponds to an equation of the form
:<math display="block">z(\mathbf{x})=z_B+\text{
By changing the entering variable choice rule so that it selects a column where the entry in the objective row is negative, the algorithm is changed so that it finds the
===Leaving variable selection===
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is the minimum over all ''r'' so that ''a''<sub>''rc''</sub> > 0. This is called the ''minimum ratio test''.<ref name="Murty66"/> If there is more than one row for which the minimum is achieved then a ''dropping variable choice rule''<ref>{{harvtxt|Murty|1983|p=67}}</ref> can be used to make the determination.
===
{{see also|Revised simplex algorithm#Numerical example}}
Consider the linear program
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:<math>
\begin{bmatrix}
0 & \frac{7}{3} & \frac{1}{3} & 0 &
0 & \frac{2}{3} & \frac{5}{3} &
\end{bmatrix}
</math>
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For the next step, there are no positive entries in the objective row and in fact
:<math display="block">Z = -20 + \frac{
so the minimum value of ''Z'' is −20.
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\end{align}</math>
It differs from the previous example by having equality instead of inequality constraints. The previous solution <math>x=y=0\, , z=5</math> violates the first constraint.
This new problem is represented by the (non-canonical) tableau
:<math>
\begin{bmatrix}
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The equation defining the original objective function is retained in anticipation of Phase II.
By construction, ''u'' and ''v'' are both
:<math>
\begin{bmatrix}
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:<math>
\begin{bmatrix}
0 & 7 & 0 & -25 & 0 & 2 & -10 & -130 \\
0 & 0 & 7 & 1 & 0 & 3 & -1 & 15 \\
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</math>
This is, fortuitously, already optimal and the optimum value for the original linear program is −130/7. This value is "worse" than -20 which is to be expected for a problem which is more constrained.
==Advanced topics==
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The tableau form used above to describe the algorithm lends itself to an immediate implementation in which the tableau is maintained as a rectangular (''m'' + 1)-by-(''m'' + ''n'' + 1) array. It is straightforward to avoid storing the m explicit columns of the identity matrix that will occur within the tableau by virtue of '''B''' being a subset of the columns of ['''A''', '''I''']. This implementation is referred to as the "''standard'' simplex algorithm". The storage and computation overhead is such that the standard simplex method is a prohibitively expensive approach to solving large linear programming problems.
In each simplex iteration, the only data required are the first row of the tableau, the (pivotal) column of the tableau corresponding to the entering variable and the right-hand-side. The latter can be updated using the pivotal column and the first row of the tableau can be updated using the (pivotal) row corresponding to the leaving variable. Both the pivotal column and pivotal row may be computed directly using the solutions of linear systems of equations involving the matrix '''B''' and a matrix-vector product using '''A'''. These observations motivate the "[[Revised simplex algorithm|''revised'' simplex algorithm]]", for which implementations are distinguished by their invertible representation of '''B'''.<ref name="DantzigThapa2" >{{cite book |first1=George B. |last1=Dantzig |authorlink=George B. Dantzig |first2=Mukund N. |last2=Thapa |year=2003 |title=Linear Programming 2: Theory and Extensions |publisher=Springer-Verlag }}</ref>
In large linear-programming problems '''A''' is typically a [[sparse matrix]] and, when the resulting sparsity of '''B''' is exploited when maintaining its invertible representation, the revised simplex algorithm is much more efficient than the standard simplex method. Commercial simplex solvers are based on the revised simplex algorithm.<ref name="Padberg" >{{cite book |first=M. |last=Padberg
===Degeneracy: stalling and cycling===
If the values of all basic variables are strictly positive, then a pivot must result in an improvement in the objective value. When this is always the case no set of basic variables occurs twice and the simplex algorithm must terminate after a finite number of steps. Basic feasible solutions where at least one of the ''basic ''variables is zero are called ''degenerate'' and may result in pivots for which there is no improvement in the objective value. In this case there is no actual change in the solution but only a change in the set of basic variables. When several such pivots occur in succession, there is no improvement; in large industrial applications, degeneracy is common and such "''stalling''" is notable.
Worse than stalling is the possibility the same set of basic variables occurs twice, in which case, the deterministic pivoting rules of the simplex algorithm will produce an infinite loop, or "cycle". While degeneracy is the rule in practice and stalling is common, cycling is rare in practice. A discussion of an example of practical cycling occurs in [[Manfred W. Padberg|Padberg]].<ref name="Padberg"/> [[Bland's rule]] prevents cycling and thus guarantees that the simplex algorithm always terminates.<ref name="Padberg"/><ref name="Bland">
{{cite journal
{{cite journal|title=New finite pivoting rules for the simplex method|first=Robert G.|last=Bland|journal=Mathematics of Operations Research|volume=2|issue=2|date=May 1977|pages=103–107|doi=10.1287/moor.2.2.103|jstor=3689647|mr=459599|s2cid=18493293|url=https://semanticscholar.org/paper/874b988e359f63c8068226c53ef0a9bcd54e5e4d}}</ref><ref>{{harvtxt|Murty|1983|p=79}}</ref> Another pivoting algorithm, the [[criss-cross algorithm]] never cycles on linear programs.<ref>There are abstract optimization problems, called [[oriented matroid]] programs, on which Bland's rule cycles (incorrectly) while the [[criss-cross algorithm]] terminates correctly.</ref>▼
| title=New finite pivoting rules for the simplex method
| first1=Robert G. | last1=Bland | authorlink1=Robert G. Bland
| journal=[[Mathematics of Operations Research]]
| volume=2
| issue=2
| date=May 1977
| pages=103–107
| doi=10.1287/moor.2.2.103
| jstor=3689647
| mr=459599
▲
History-based pivot rules such as [[Zadeh's rule]] and [[Cunningham's rule]] also try to circumvent the issue of stalling and cycling by keeping track of how often particular variables are being used and then favor such variables that have been used least often.
===Efficiency in the worst case===
The simplex method is remarkably efficient in practice and was a great improvement over earlier methods such as [[Fourier–Motzkin elimination]]. However, in 1972, [[Victor Klee|Klee]] and Minty<ref name="KleeMinty">{{cite book|title=Inequalities III (Proceedings of the Third Symposium on Inequalities held at the University of California, Los Angeles, Calif., September 1–9, 1969, dedicated to the memory of Theodore S. Motzkin)|editor-first=Oved|editor-last=Shisha|publisher=Academic Press|___location=New York-London|year=1972|mr=332165|last1=Klee|first1=Victor|author-link1=Victor Klee|last2=Minty|first2= George J.|author-link2=George J. Minty|chapter=How good is the simplex algorithm?|pages=159–175}}</ref> gave an example, the [[Klee–Minty cube]], showing that the worst-case complexity of simplex method as formulated by Dantzig is [[exponential time]]. Since then, for almost every variation on the method, it has been shown that there is a family of linear programs for which it performs badly. It is an open question if there is a variation with [[polynomial time]], although sub-exponential pivot rules are known.<ref>{{Citation
| last1 = Hansen
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| last2 = Zwick
| first2 = Uri
|
| author2-link = Uri Zwick
▲ | title = An Improved Version of the Random-Facet Pivoting Rule for the Simplex Algorithm
▲ | journal = Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing
| pages = 209–218
| year = 2015
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</ref>
In 2014, it was proved{{citation-needed|date=January 2024}} that a particular variant of the simplex method is [[NP-mighty]], i.e., it can be used to solve, with polynomial overhead, any problem in NP implicitly during the algorithm's execution. Moreover, deciding whether a given variable ever enters the basis during the algorithm's execution on a given input, and determining the number of iterations needed for solving a given problem, are both [[NP-hardness|NP-hard]] problems.<ref>{{Cite journal|last1=Disser|first1=Yann|last2=Skutella|first2=Martin|date=2018-11-01|title=The Simplex Algorithm Is NP-Mighty|journal=ACM Trans. Algorithms|volume=15|issue=1|pages=5:1–5:19|doi=10.1145/3280847|issn=1549-6325|arxiv=1311.5935|s2cid=54445546}}</ref> At about the same time it was shown that there exists an artificial pivot rule for which computing its output is [[PSPACE-complete]].<ref>{{Citation | last1 = Adler | first1 = Ilan|author1-link=Ilan Adler | last2 = Christos | first2 = Papadimitriou | author2-link = Christos Papadimitriou | last3 = Rubinstein | first3 = Aviad | title =
===Efficiency in practice===
Analyzing and quantifying the observation that the simplex algorithm is efficient in practice despite its exponential worst-case complexity has led to the development of other measures of complexity. The simplex algorithm has polynomial-time [[Best, worst and average case|average-case complexity]] under various [[probability distribution]]s, with the precise average-case performance of the simplex algorithm depending on the choice of a probability distribution for the [[random matrix|random matrices]].<ref name="Schrijver"
Another method to analyze the performance of the simplex algorithm studies the behavior of worst-case scenarios under small perturbation – are worst-case scenarios stable under a small change (in the sense of [[structural stability]]), or do they become tractable? This area of research, called [[smoothed analysis]], was introduced specifically to study the simplex method. Indeed, the running time of the simplex method on input with noise is polynomial in the number of variables and the magnitude of the perturbations.<ref>{{Cite book | last1=Spielman | first1=Daniel | last2=Teng | first2=Shang-Hua | author2-link=Shanghua Teng | title=Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing | publisher=ACM | isbn=978-1-58113-349-3 | doi=10.1145/380752.380813 | year=2001 | chapter=Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time| pages=296–305 | arxiv=cs/0111050| s2cid=1471 }}</ref><ref>{{Cite journal|last1=Dadush|first1=Daniel|last2=Huiberts|first2=Sophie|date=2020-01-01|title=A Friendly Smoothed Analysis of the Simplex Method|url=https://epubs.siam.org/doi/abs/10.1137/18M1197205|journal=SIAM Journal on Computing|volume=49|issue=5|pages=STOC18–449|doi=10.1137/18M1197205|s2cid=226351624|issn=0097-5397|arxiv=1711.05667}}</ref>
==Other algorithms==
Other algorithms for solving linear-programming problems are described in the [[linear programming|linear-programming]] article. Another basis-exchange pivoting algorithm is the [[criss-cross algorithm]].<ref>{{cite journal|last1=Terlaky|first1=Tamás|last2=Zhang|first2=Shu Zhong|title=Pivot rules for linear programming: A Survey on recent theoretical developments|issue=1|journal=Annals of Operations Research|volume=46–47|year=1993|pages=203–233|doi=10.1007/BF02096264|mr=1260019|citeseerx = 10.1.1.36.7658 |s2cid=6058077|issn=0254-5330}}</ref><ref>{{cite
==Linear-fractional programming==
{{Main|Linear-fractional programming}}
[[Linear-fractional programming|Linear–fractional programming]] (LFP) is a generalization of [[linear programming]] (LP). In LP the objective function is a [[linear functional|linear function]], while the objective function of a linear–fractional program is a ratio of two linear functions. In other words, a linear program is a fractional–linear program in which the denominator is the constant function having the value one everywhere. A linear–fractional program can be solved by a variant of the simplex algorithm<ref>{{harvtxt|Murty|1983|loc=Chapter 3.20 (pp. 160–164) and pp. 168 and 179}}</ref><ref>Chapter five: {{cite book|last=Craven|first=B. D.|title=Fractional programming|series=Sigma Series in Applied Mathematics|volume=4|publisher=Heldermann Verlag|___location=Berlin|year=1988|pages=145|isbn=978-3-88538-404-5|mr=949209}}</ref><ref>{{cite journal|last1=Kruk|first1=Serge|last2=Wolkowicz|first2=Henry|title=Pseudolinear programming|journal=[[SIAM Review]]|volume=41|year=1999|issue=4|pages=795–805|mr=1723002|jstor=2653207|doi=10.1137/S0036144598335259|citeseerx=10.1.1.53.7355|bibcode=1999SIAMR..41..795K}}
</ref><ref>{{cite journal|last1=Mathis|first1=Frank H.|last2=Mathis|first2=Lenora Jane|title=A nonlinear programming algorithm for hospital management|journal=[[SIAM Review]]|volume=37 |year=1995 |issue=2 |pages=230–234|mr=1343214|jstor=2132826|doi=10.1137/1037046|s2cid=120626738 }}
</ref> or by the [[criss-cross algorithm]].<ref>{{cite journal|title=The finite criss-cross method for hyperbolic programming|journal=European Journal of Operational Research|volume=114|issue=1|
pages=198–214|year=1999|issn=0377-2217|doi=10.1016/S0377-2217(98)00049-6|first1=Tibor|last1=Illés|first2=Ákos|last2=Szirmai|first3=Tamás|last3=Terlaky|url=http://www.cas.mcmaster.ca/~terlaky/files/dut-twi-96-103.ps.gz |citeseerx=10.1.1.36.7090}}</ref>
==
{{div col}}
* [[Bland's rule|Pivoting rule of Bland]], which avoids cycling▼
* [[Criss-cross algorithm]]
* [[Cutting-plane method]]
* [[Devex algorithm]]
* [[Fourier–Motzkin elimination]]
* [[Gradient descent]]
* [[Karmarkar's algorithm]]
* [[Nelder–Mead method|Nelder–Mead simplicial heuristic]]
* [[Loss Functions]] - a type of Objective Function
▲* [[Bland's rule|Pivoting rule of Bland]], which avoids cycling
{{colend}}
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* {{cite book|last=Murty|first=Katta G.|author-link=Katta G. Murty|title=Linear programming|publisher=John Wiley & Sons, Inc.|___location=New York|year=1983|pages=xix+482|isbn=978-0-471-09725-9|mr=720547}}
==
These introductions are written for students of [[computer science]] and [[operations research]]:
* [[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.
* Frederick S. Hillier and Gerald J. Lieberman: ''Introduction to Operations Research'', 8th edition. McGraw-Hill. {{isbn|0-07-123828-X}}
* {{cite book|title=Optimization in operations research|first=Ronald L.|last=Rardin|year=1997|publisher=Prentice Hall|pages=919|isbn=978-0-02-398415-0}}
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==External links==
{{wikibooks|Operations Research|The Simplex Method}}
* [http://www.isye.gatech.edu/~spyros/LP/LP.html An Introduction to Linear Programming and the Simplex Algorithm] by Spyros Reveliotis of the Georgia Institute of Technology.
* Greenberg, Harvey J., ''Klee–Minty Polytope Shows Exponential Time Complexity of Simplex Method'' the University of Colorado at Denver (1997) [http://glossary.computing.society.informs.org/notes/Klee-Minty.pdf PDF download]
* [http://www.lokminglui.com/lpch3.pdf Simplex Method] A tutorial for Simplex Method with examples (also two-phase and M-method).
* [https://www.mathstools.com/section/main/simplex_online_calculator Mathstools] Simplex Calculator from www.mathstools.com
* [http://math.uww.edu/~mcfarlat/s-prob.htm Example of Simplex Procedure for a Standard Linear Programming Problem] by Thomas McFarland of the University of Wisconsin-Whitewater.
* [http://www.phpsimplex.com/simplex/simplex.htm?l=en PHPSimplex: online tool to solve Linear Programming Problems] by Daniel Izquierdo and Juan José Ruiz of the University of Málaga (UMA, Spain)
* [
{{Optimization algorithms|convex}}
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