Simplex algorithm: Difference between revisions

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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 Inc.1, 2000 |url=http://www.computer.org/csdl/mags/cs/2000/01/c1022.html}}</ref>{{failed verification|date=June 2025|reason=Could not locate reference.}}

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.

== History ==

[[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#Education|he had mistaken]] as homework in his professor [[Jerzy Neyman]]'s class (and actually later solved), was applicable to finding an algorithm for linear programs. This problem involved finding the existence of [[Lagrange multipliers on Banach spaces|Lagrange multipliers]] for general linear programs over a continuum of variables, each bounded between zero and one, and satisfying linear constraints expressed in the form of [[Lebesgue integral]]s. Dantzig later published his "homework" as a thesis to earn his doctorate. The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient.<ref>{{Cite encyclopedia|url = http://apps.dtic.mil/dtic/tr/fulltext/u2/a182708.pdf|archive-url = https://web.archive.org/web/20150529003047/http://www.dtic.mil/dtic/tr/fulltext/u2/a182708.pdf|url-status = live|archive-date = May 29, 2015|title = Origins of the simplex method|last = Dantzig|first = George|date = May 1987|workencyclopedia = A History of Scientific Computing|editor-last=Nash|editor-first=Stephen G.|publisher=Association for Computing Machinery|pages = 141–151|doi = 10.1145/87252.88081|isbn = 978-0-201-50814-7}}</ref>

== 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

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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 i \ x_i \ge 0</math>

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 ''<math>\mathbf{b''}</math> (different authors use different conventions as to the exact layout). If the columns of <math>\mathbf{A}</math> can be rearranged so that it contains the [[identity matrix]] of order ''<math>p''</math> (the number of rows in <math>\mathbf{A}</math>) then the tableau is said to be in ''canonical form''.<ref>{{harvtxt|Murty|1983|loc=section 2.3.2}}</ref> The variables corresponding to the columns of the identity matrix are called ''basic variables'' while the remaining variables are called ''nonbasic'' or ''free variables''. If the values of the nonbasic variables are set to 0, then the values of the basic variables are easily obtained as entries in ''<math>\mathbf{b''}</math> and this solution is a basic feasible solution. The algebraic interpretation here is that the coefficients of the linear equation represented by each row are either <math>0</math>, <math>1</math>, or some other number. Each row will have <math>1</math> column with value <math>1</math>, <math>p-1</math> columns with coefficients <math>0</math>, and the remaining columns with some other coefficients (these other variables represent our non-basic variables). By setting the values of the non-basic variables to zero we ensure in each row that the value of the variable represented by a <math>1</math> in its column is equal to the <math>b</math> value at that row.

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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is the minimum over all ''r'' so that ''a''_''rc'' > 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.

=== Example ===

{{see also|Revised simplex algorithm#Numerical example}}

Consider the linear program

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:<math>

\begin{bmatrix}

31 & -\frac{2}{3} & -\frac{11}{3} & 0 & 0 & -\frac{4}{3} & -6020 \\

0 & \frac{7}{3} & \frac{1}{3} & 0 & 31 & -\frac{1}{3} & 15 5 \\

0 & \frac{2}{3} & \frac{5}{3} & 31 & 0 & \frac{1}{3} & 155

\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{-60+2x+11y+4t2}{3} = -20 x+ \frac{2x11}{3}y+11y+4t\frac{4}{3}t</math>

so the minimum value of ''Z'' is&nbsp;&minus;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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</math>

This is, fortuitously, already optimal and the optimum value for the original linear program is&nbsp;−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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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&nbsp;'''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 |title=Linear Optimization and Extensions |edition=Second |publisher=Springer-Verlag |year=1999 |isbn=3-540-65833-5 }}</ref><ref name="DantzigThapa2"/><ref>{{cite book |first1=Dmitris |last1=Alevras |first2=Manfred W. |last2=Padberg |title=Linear Optimization and Extensions: Problems and Solutions |series=Universitext |publisher=Springer-Verlag |year=2001 |isbn=3-540-41744-3 }} (Problems from Padberg with solutions.)</ref><ref name="MarosMitra" >{{cite book|last1=Maros|first1=István|last2=Mitra|author2-link=Gautam Mitra|first2=Gautam|chapter=Simplex algorithms|mr=1438309|title=Advances in linear and integer programming|pages=1–46|editor=J. E. Beasley|publisher=Oxford Science|year=1996}}</ref><ref>{{cite book|mr=1960274|last=Maros|first=István|title=Computational techniques of the simplex method|series=International Series in Operations Research & Management Science|volume=61|publisher=Kluwer Academic Publishers|___location=Boston, MA|year=2003|pages=xx+325|isbn=978-1-4020-7332-8}}</ref>

===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 20151977

| pages=103–107

| doi=10.1287/moor.2.2.103

| jstor=3689647

| mr=459599

{{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>

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.

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| title = Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

| chapter = An Improved Version of the Random-Facet Pivoting Rule for the Simplex Algorithm

| date = 2015

| author2-link = Uri Zwick

| pages = 209–218

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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 = Integer Programming and Combinatorial Optimization | chapter = On Simplex Pivoting Rules and Complexity Theory | volume = 17 | pages = 13–24 | year = 2014 | arxiv = 1404.3320 | doi = 10.1007/978-3-319-07557-0_2| series = Lecture Notes in Computer Science | isbn = 978-3-319-07556-3 | s2cid = 891022 }}</ref> In 2015, this was strengthened to show that computing the output of Dantzig's pivot rule is [[PSPACE-complete]].<ref>{{Citation | last1 = Fearnly | first1 = John | last2 = Savani | first2 = Rahul | title = Proceedings of the forty-seventh annual ACM symposium on Theory of Computing | chapter = The Complexity of the Simplex Method | date = 2015 | pages = 201–208 | year = 2015 | arxiv = 1404.0605 | doi = 10.1145/2746539.2746558| isbn = 9781450335362 | s2cid = 2116116 }}</ref>

=== 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">[[Alexander Schrijver]], ''Theory of Linear and Integer Programming''. John Wiley & sons, 1998, {{isbn|0-471-98232-6}} (mathematical)</ref><ref name="Borgwardt">The simplex algorithm takes on average ''D'' steps for a cube. {{harvtxt|Borgwardt|1987}}: {{cite book|last=Borgwardt|first=Karl-Heinz|title=The simplex method: A probabilistic analysis|series=Algorithms and Combinatorics (Study and Research Texts)|volume=1|publisher=Springer-Verlag|___location=Berlin|year=1987|pages=xii+268|isbn=978-3-540-17096-9|mr=868467}}</ref> Another approach to studying "[[porous set|typical phenomena]]" uses [[Baire category theory]] from [[general topology]], and to show that (topologically) "most" matrices can be solved by the simplex algorithm in a polynomial number of steps.{{Citation needed|date=June 2019}}

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 journal|first1=Komei|last1=Fukuda|author1-link=Komei Fukuda|first2=Tamás|last2=Terlaky|author2-link=Tamás Terlaky|title=Criss-cross methods: A fresh view on pivot algorithms |journal=Mathematical Programming, Series B|volume=79|number=1–3|pages=369–395|editor1=Thomas M. Liebling |editor2=Dominique de Werra|publisher=North-Holland Publishing |___location=Amsterdam|year=1997|doi=10.1007/BF02614325|mr=1464775|s2cid=2794181 |url=http://infoscience.epfl.ch/record/77270 }}</ref> There are polynomial-time algorithms for linear programming that use interior point methods: these include [[Khachiyan]]'s [[ellipsoidal algorithm]], [[Karmarkar]]'s [[Karmarkar's algorithm|projective algorithm]], and [[interior point method|path-following algorithm]]s.<ref name="Vanderbei"/> The [[Big_M_method|Big-M method]] is an alternative strategy for solving a linear program, using a single-phase simplex.

==Linear-fractional programming==

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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>

== See also ==

{{div col}}

* [[Bland's rule|Pivoting rule of Bland]], which avoids cycling

* [[Criss-cross algorithm]]

* [[Cutting-plane method]]

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* [[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}}

== Further reading ==

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.&nbsp;790&ndash;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)

* [httphttps://wwwweb.archive.org/web/20180513100217/http://simplex-m.com/ simplex-m] Online Simplex Solver

{{Optimization algorithms|convex}}