Linear programming: Difference between revisions

Revision as of 08:37, 28 March 2021

Latest revision as of 21:36, 9 August 2025

(149 intermediate revisions by 91 users not shown)

Line 1:

{{short description|Method to solve some optimization problems}}

{{for|the retronym referring to television broadcasting|Broadcast programming}}

[[File:Linear optimization in a 2-dimensional polytope.svg|thumb|A pictorial representation of a simple linear program with two variables and six inequalities. The set of feasible solutions is depicted in yellow and forms a [[polygon]], a 2-dimensional [[polytope]]. The optimum of the linear cost function is represented bywhere the red line andintersects the arrow:polygon. The red line is a [[level set]] of the cost function, and the arrow indicates the direction in which we are optimizing.]]

[[File:3dpoly.svg|thumb|right|A closed feasible region of a problem with three variables is a convex [[polyhedron]]. The surfaces giving a fixed value of the objective function are [[Plane (geometry)|planes]] (not shown). The linear programming problem is to find a point on the polyhedron that is on the plane with the highest possible value.]]

'''Linear programming''' ('''LP'''), also called '''linear optimization'''), is a method to achieve the best outcome (such as maximum profit or lowest cost) in a [[mathematical model]] whose requirements and objective are represented by [[linear function#As a polynomial function|linear relationships]]. Linear programming is a special case of mathematical programming (also known as [[mathematical optimization]]).

More formally, linear programming is a technique for the [[mathematical optimization|optimization]] of a [[linear]] [[objective function]], subject to [[linear equality]] and [[linear inequality]] [[Constraint (mathematics)|constraints]]. Its [[feasible region]] is a [[convex polytope]], which is a set defined as the [[intersection (mathematics)|intersection]] of finitely many [[Half-space (geometry)|half spaces]], each of which is defined by a linear inequality<!-- ; alternatively, a convex polytope is the [[Minkowski sum]] of a [[convex polytope]] and a convex [[polyhedral cone]] -->. Its objective function is a [[real number|real]]-valued [[affine function|affine (linear) function]] defined on this polyhedronpolytope. A linear programming [[algorithm]] finds a point in the [[polytope]] where this function has the smallestlargest (or largestsmallest) value if such a point exists.

Linear programs are problems that can be expressed in [[canonical form|standard form]] as:

:<math> \begin{align}

& \text{Find a vector} && \mathbf{x} \\

& \text{that maximizes} && \mathbf{c}^\mathsf{T} \mathbf{x}\\

& \text{subject to} && A \mathbf{x} \leqle \mathbf{b} \\

& \text{and} && \mathbf{x} \ge \mathbf{0}.

\end{align} </math>

Here the components of '''<math>\mathbf{x'''}</math> are the variables to be determined, '''<math>\mathbf{c'''}</math> and '''<math>\mathbf{b'''}</math> are given [[vector space|vectors]], (withand <math>\mathbf{c}^TA</math> indicating that the coefficients of '''c''' are used as a single-row matrix for the purpose of forming the matrix product), and ''A'' is a given [[Matrix (mathematics)|matrix]]. The function whose value is to be maximized or minimized (<math>\mathbf x\mapsto\mathbf{c}^\mathsf{T}\mathbf{x}</math> in this case) is called the [[objective function]]. The inequalitiesconstraints ''<math>A''''' \mathbf{x'''&nbsp;≤&nbsp;'''} \le \mathbf{b'''}</math> and '''<math>\mathbf{x'''} ≥\geq '''\mathbf{0''' are the constraints which}</math> specify a [[convex polytope]] over which the objective function is to be optimized. In this context, two vectors are [[Comparability|comparable]] when they have the same dimensions. If every entry in the first is less-than or equal-to the corresponding entry in the second, then it can be said that the first vector is less-than or equal-to the second vector.

Linear programming can be applied to various fields of study. It is widely used in mathematics, and, to a lesser extent, in business, [[economics]], and for some engineering problems. IndustriesThere thatis usea linearclose programmingconnection modelsbetween includelinear transportationprograms, energyeigenequations, telecommunications[[John von Neumann]]'s general equilibrium model, and manufacturing.structural Itequilibrium hasmodels proven(see useful[[dual inlinear modelingprogram]] diversefor typesdetails).<ref>{{cite ofjournal problems|last=von inNeumann [[automated|first=J. planning|year=1945 and scheduling|planning]],title=A [[routing]],Model [[schedulingof (productionGeneral Economic Equilibrium processes)|scheduling]],journal=The [[assignmentReview of Economic Studies problem|assignment]],volume=13 and|issue=1 design|pages=1–9|doi=10.2307/2296111 |jstor=2296111 }}</ref>

<ref>{{Cite journal

| last1 = Kemeny

| first1 = J. G.

| last2 = Morgenstern

| first2 = O.

| last3 = Thompson

| first3 = G. L.

| year = 1956

| title = A Generalization of the von Neumann Model of an Expanding Economy

| journal = Econometrica

| volume = 24

| issue = 2

| pages = 115–135

| doi = 10.2307/1905746

| jstor = 1905746

}}</ref>

<ref>{{Cite book

| last = Li

| first = Wu

| title = General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics

| year = 2019

| publisher = Economic Science Press

| ___location = Beijing

| isbn = 978-7-5218-0422-5

| language = zh

| pages = 122–125

}}</ref>

Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proven useful in modeling diverse types of problems in [[automated planning and scheduling|planning]], [[routing]], [[scheduling (production processes)|scheduling]], [[assignment problem|assignment]], and design.

== History ==

Line 25 ⟶ 53:

The problem of solving a system of linear inequalities dates back at least as far as [[Joseph Fourier|Fourier]], who in 1827 published a method for solving them,<ref name="SierksmaZwols2015">{{cite book|author1=Gerard Sierksma|author2=Yori Zwols|title=Linear and Integer Optimization: Theory and Practice|edition=3rd|year=2015|publisher=CRC Press|isbn=978-1498710169|page=1}}</ref> and after whom the method of [[Fourier–Motzkin elimination]] is named.

In the late 1930s, Soviet mathematician [[Leonid Kantorovich]] and American economist [[Wassily Leontief]] independently delved into the practical applications of linear programming. Kantorovich focused on manufacturing schedules, while Leontief explored economic applications. Their groundbreaking work was largely overlooked for decades.

In 1939 a linear programming formulation of a problem that is equivalent to the general linear programming problem was given by the [[Soviet Union|Soviet]] [[mathematician]] and [[economist]] [[Leonid Kantorovich]], who also proposed a method for solving it.<ref name="Schrijver1998">{{cite book|author=Alexander Schrijver|title=Theory of Linear and Integer Programming|year=1998|publisher=John Wiley & Sons|isbn=978-0-471-98232-6|pages=221–222}}</ref> It is a way he developed, during [[World War II]], to plan expenditures and returns in order to reduce costs of the army and to increase losses imposed on the enemy.{{Citation needed|date=August 2017}} Kantorovich's work was initially neglected in the [[USSR]].<ref name="dantzig1982">{{cite journal|url = http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA112060|title = Reminiscences about the origins of linear programming|author=George B. Dantzig|date = April 1982|journal = Operations Research Letters|volume = 1|issue = 2|pages = 43–48|doi = 10.1016/0167-6377(82)90043-8}}</ref> About the same time as Kantorovich, the Dutch-American economist [[Tjalling Koopmans|T. C. Koopmans]] formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 [[Nobel prize in economics]].<ref name="SierksmaZwols2015" /> In 1941, [[Frank Lauren Hitchcock]] also formulated transportation problems as linear programs and gave a solution very similar to the later [[simplex method]].<ref name="Schrijver1998" /> Hitchcock had died in 1957 and the Nobel prize is not awarded posthumously.

The turning point came during World War II when linear programming emerged as a vital tool. It found extensive use in addressing complex wartime challenges, including transportation logistics, scheduling, and resource allocation. Linear programming proved invaluable in optimizing these processes while considering critical constraints such as costs and resource availability.

During 1946–1947, [[George Dantzig|George B. Dantzig]] independently developed general linear programming formulation to use for planning problems in the US Air Force.<ref name=":0">{{Cite book|title=Linear programming|last1=Dantzig|first1=George B.|last2=Thapa|first2=Mukund Narain|date=1997|publisher=Springer|isbn=0387948333|___location=New York|page=xxvii|oclc=35318475}}</ref> In 1947, Dantzig also invented the [[Simplex algorithm|simplex method]] that for the first time efficiently tackled the linear programming problem in most cases.<ref name=":0" /> When Dantzig arranged a meeting with [[John von Neumann]] to discuss his simplex method, Neumann immediately conjectured the theory of [[#Duality|duality]] by realizing that the problem he had been working in [[game theory]] was equivalent.<ref name=":0" /> Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948.<ref name="dantzig1982"/> Dantzig's work was made available to public in 1951. In the post-war years, many industries applied it in their daily planning.

Despite its initial obscurity, the wartime successes propelled linear programming into the spotlight. Post-WWII, the method gained widespread recognition and became a cornerstone in various fields, from operations research to economics. The overlooked contributions of Kantorovich and Leontief in the late 1930s eventually became foundational to the broader acceptance and utilization of linear programming in optimizing decision-making processes.<ref>{{Cite web |title=Linear programming {{!}} Definition & Facts {{!}} Britannica |url=https://www.britannica.com/science/linear-programming-mathematics |access-date=2023-11-20 |website=www.britannica.com |language=en}}</ref>

In 1939 a linear programming formulation of a problem that is equivalent to the general linear programming problem was given by the [[Soviet Union|Soviet]] [[mathematician]] and [[economist]] [[Leonid Kantorovich]], who also proposed a method for solving it.<ref name="Schrijver1998">{{cite book|author=Alexander Schrijver|title=Theory of Linear and Integer Programming|year=1998|publisher=John Wiley & Sons|isbn=978-0-471-98232-6|pages=221–222}}</ref> It is a way he developed, during [[World War II]], to plan expenditures and returns in order to reduce costs of the army and to increase losses imposed on the enemy.{{Citation needed|date=August 2017}} Kantorovich's work was initially neglected in the [[USSR]].<ref name="dantzig1982">{{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|author=George B. Dantzig|date = April 1982|journal = Operations Research Letters|volume = 1|issue = 2|pages = 43–48|doi = 10.1016/0167-6377(82)90043-8}}</ref> About the same time as Kantorovich, the Dutch-American economist [[Tjalling Koopmans|T. C. Koopmans]] formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 [[Nobel prizeMemorial Prize in economicsEconomic Sciences]].<ref name="SierksmaZwols2015" /> In 1941, [[Frank Lauren Hitchcock]] also formulated transportation problems as linear programs and gave a solution very similar to the later [[simplex method]].<ref name="Schrijver1998">{{cite book |author=Alexander Schrijver |title=Theory of Linear and Integer Programming |publisher=John Wiley & Sons |year=1998 |isbn=978-0-471-98232-6 |pages=221–222}}</ref> Hitchcock had died in 1957, and the Nobel prizeMemorial Prize is not awarded posthumously.

DuringFrom 1946–1947,1946 to 1947 [[George Dantzig|George B. Dantzig]] independently developed general linear programming formulation to use for planning problems in the US Air Force.<ref name=":0">{{Cite book|title=Linear programming|last1=Dantzig|first1=George B.|last2=Thapa|first2=Mukund Narain|date=1997|publisher=Springer|isbn=0387948333|___location=New York|page=xxvii|oclc=35318475}}</ref> In 1947, Dantzig also invented the [[Simplex algorithm|simplex method]] that, for the first time efficiently, tackled the linear programming problem in most cases.<ref name=":0" /> When Dantzig arranged a meeting with [[John von Neumann]] to discuss his simplex method, von Neumann immediately conjectured the theory of [[#Duality|duality]] by realizing that the problem he had been working in [[game theory]] was equivalent.<ref name=":0" /> Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948.<ref name="dantzig1982"/> Dantzig's work was made available to public in 1951. In the post-war years, many industries applied it in their daily planning.

Dantzig's original example was to find the best assignment of 70 people to 70 jobs. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the [[Abundance of the chemical elements|number of particles]] in the [[observable universe]]. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the [[simplex algorithm]]. The theory behind linear programming drastically reduces the number of possible solutions that must be checked.

Line 34 ⟶ 68:

== Uses ==

Linear programming is a widely used field of optimization for several reasons. Many practical problems in [[operations research]] can be expressed as linear programming problems.<ref name="dantzig1982"/> Certain special cases of linear programming, such as ''[[network flow problem|network flow]]'' problems and [[multi-commodity flow problem|''multicommodity flow'' problems]], are considered important enough to have generated much research on specialized algorithms for their solution. A number of algorithms for other types of optimization problems work by solving LPlinear programming problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as ''duality,'' ''decomposition,'' and the importance of ''convexity'' and its generalizations. Likewise, linear programming was heavily used in the early formation of [[microeconomics]], and it is currently utilized in company management, such as planning, production, transportation, technology and other issuestechnology. Although the modern management issues are ever-changing, most companies would like to [[profit maximization|maximize profits]] and minimize costs with limited resources. Therefore,Google manyalso issuesuses canlinear beprogramming characterizedto asstabilize linearYouTube programmingvideos.<ref>{{cite problemsbook |author1=M. Grundmann |author2=V. Kwatra |author3=I. Essa |title=CVPR 2011 |chapter=Auto-directed video stabilization with robust L1 optimal camera paths |date=2011 |pages=225–232 |doi=10.1109/CVPR.2011.5995525 |isbn=978-1-4577-0394-2 |s2cid=17707171 |url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/37041.pdf |language=English}}</ref>

== Standard form ==

''Standard form'' is the usual and most intuitive form of describing a linear programming problem. It consists of the following three parts:

* A '''linear (or affine) function to be maximized'''

: e.g. <math> f(x_{1},x_{2}) = c_1 x_1 + c_2 x_2</math>

* '''Problem constraints''' of the following form

Line 55 ⟶ 89:

The problem is usually expressed in ''[[Matrix (mathematics)|matrix]] form'', and then becomes:

: <math>\max \{\, \mathbf{c}^\mathrmmathsf{T} \mathbf{x} \;|mid \;mathbf{x}\in\mathbb{R}^n\land A \mathbf{x} \leq \mathbf{b} \land \mathbf{x} \geq 0 \,\}</math>

Other forms, such as minimization problems, problems with constraints on alternative forms, as well asand problems involving negative [[variable (programming)|variables]] can always be rewritten into an equivalent problem in standard form.

=== Example ===

[[File:linear_programming_feasible_region_farmer_example.svg|thumb|Graphical solution to the farmer example &ndash; after shading regions violating the conditions, the vertex of the unshaded region with the dashed line farthest from the origin gives the optimal combination (its lying on the land and pesticide lines implies that revenue is limited by land and pesticide, not fertilizer)]]

Suppose that a farmer has a piece of farm land, say ''L'' km²[[hectare]]s, to be planted with either wheat or barley or some combination of the two. The farmer has a limited amount of fertilizer, ''F'' kilograms, of fertilizer and pesticide, ''P'' kilograms of pesticide. Every square kilometerhectare of wheat requires ''F''₁ kilograms of fertilizer and ''P''₁ kilograms of pesticide, while every square kilometerhectare of barley requires ''F''₂ kilograms of fertilizer and ''P''₂ kilograms of pesticide. Let S₁ be the selling price of wheat per square kilometer, and S₂ be the selling price of barley, per hectare. If we denote the area of land planted with wheat and barley by ''x''₁ and ''x''₂ respectively, then profit can be maximized by choosing optimal values for ''x''₁ and ''x''₂. This problem can be expressed with the following linear programming problem in the standard form:

{|

|-

| colspanvalign="2top" | Maximize:
| valign="top"|<math>S_1\cdot x_1+S_2\cdot x_2</math>

| (maximize the revenue (the total wheat sales plus the total barley sales) – revenue is the "objective function")

|-

| {{nowrap|Subject to:}}

| <math>x_1 + x_2\leq L</math>

| (limit on total area)

Line 92 ⟶ 128:

: <math>

\begin{bmatrix}

1 & -\mathbf{c}^\mathsf{T} & 0 \\

0 & \mathbf{A} & \mathbf{I}

\end{bmatrix}

Line 131 ⟶ 167:

In matrix form this becomes:

: Maximize <math>z</math>:

: <math display=block>

\begin{bmatrix}

1 & -S_1 & -S_2 & 0 & 0 & 0 \\

Line 202 ⟶ 238:

=== Existence of optimal solutions ===

Geometrically, the linear constraints define the [[feasible region]], which is a [[convex set|convex]] [[polyhedronpolytope]]. A [[linear functional|linear function]] is a [[convex function]], which implies that every [[local minimum]] is a [[global minimum]]; similarly, a linear function is a [[concave function]], which implies that every [[local maximum]] is a [[global maximum]].

An optimal solution need not exist, for two reasons. First, if the constraints are inconsistent, then no feasible solution exists: For instance, the constraints '''x'''&nbsp;≥&nbsp;2 and '''x'''&nbsp;≤&nbsp;1 cannot be satisfied jointly; in this case, we say that the LP is ''infeasible''. Second, when the [[polytope]] is unbounded in the direction of the gradient of the objective function (where the gradient of the objective function is the vector of the coefficients of the objective function), then no optimal value is attained because it is always possible to do better than any finite value of the objective function.

=== Optimal vertices (and rays) of polyhedra ===

Otherwise, if a feasible solution exists and if the constraint set is bounded, then the optimum value is always attained on the boundary of the constraint set, by the ''[[maximum principle]]'' for ''[[convex function]]s'' (alternatively, by the ''minimum'' principle for ''[[concave function]]s'') since linear functions are both convex and concave. However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (that isi.e., the constant function taking the value zero everywhere). For this feasibility problem with the zero-function for its objective-function, if there are two distinct solutions, then every convex combination of the solutions is a solution.

The vertices of the polytope are also called ''basic feasible solutions''. The reason for this choice of name is as follows. Let ''d'' denote the number of variables. Then the fundamental theorem of linear inequalities implies (for feasible problems) that for every vertex '''x'''^* of the LP feasible region, there exists a set of ''d'' (or fewer) inequality constraints from the LP such that, when we treat those ''d'' constraints as equalities, the unique solution is '''x'''^*. Thereby we can study these vertices by means of looking at certain subsets of the set of all constraints (a discrete set), rather than the continuum of LP solutions. This principle underlies the [[simplex algorithm]] for solving linear programs.

Line 220 ⟶ 256:

==== Simplex algorithm of Dantzig ====

The [[simplex algorithm]], developed by [[George Dantzig]] in 1947, solves LP problems by constructing a feasible solution at a vertex of the [[polytope]] and then walking along a path on the edges of the polytope to vertices with non-decreasing values of the objective function until an optimum is reached for sure. In many practical problems, "[[Simplex algorithm#Degeneracy: stalling and cycling|stalling]]" occurs: many pivots are made with no increase in the objective function.<ref name="DT03">{{harvtxt|Dantzig|Thapa|2003}}</ref><ref name="Padberg">{{harvtxt|Padberg|1999}}</ref> In rare practical problems, the usual versions of the simplex algorithm may actually "cycle".<ref name="Padberg" /> To avoid cycles, researchers developed new pivoting rules.<ref name="Bland">{{harvtxt|Bland|1977}}</ref><ref name="Murty">{{harvtxt|Murty|1983}}</ref><ref name=" DT03" /><ref name=" Padberg" /><ref name="PS">{{harvtxt|Papadimitriou|Steiglitz|}}</ref><ref name="FukudaTerlaky" />

In practice, the simplex [[algorithm]] is quite efficient and can be guaranteed to find the global optimum if certain precautions against ''cycling'' are taken. The simplex algorithm has been proved to solve "random" problems efficiently, i.e. in a cubic number of steps,<ref>{{harvtxt|Borgwardt|1987}}</ref> which is similar to its behavior on practical problems.<ref name="DT03" /><ref name="Todd">{{harvtxt|Todd|2002}}</ref>

However, the simplex algorithm has poor worst-case behavior: Klee and Minty constructed a family of linear programming problems for which the simplex method takes a number of steps exponential in the problem size.<ref name="DT03" /><ref name="Murty" >{{harvtxt|Murty|1983}}</ref><ref name="PS " >{{harvtxt|Papadimitriou|Steiglitz|}}</ref> In fact, for some time it was not known whether the linear programming problem was solvable in [[polynomial time]], i.e. of [[P (complexity)|complexity class P]].

==== Criss-cross algorithm ====

Like the simplex algorithm of Dantzig, the [[criss-cross algorithm]] is a basis-exchange algorithm that pivots between bases. However, the criss-cross algorithm need not maintain feasibility, but can pivot rather from a feasible basis to an infeasible basis. The criss-cross algorithm does not have [[time complexity|polynomial time-complexity]] for linear programming. Both algorithms visit all&nbsp;2^''D'' corners of a (perturbed) [[unit cube|cube]] in dimension&nbsp;''D'', the [[Klee–Minty cube]], in the [[worst-case complexity|worst case]].<ref name="FukudaTerlaky">{{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|editor=Thomas M. Liebling |editor2=Dominique de Werra|year=1997|doi=10.1007/BF02614325|mr=1464775|citeseerx=10.1.1.36.9373|s2cid=2794181}}</ref><ref name="Roos">{{cite journal|last=Roos|first=C.|title=An exponential example for Terlaky's pivoting rule for the criss-cross simplex method|journal=Mathematical Programming|volume=46|year=1990|series=Series A|doi=10.1007/BF01585729|mr=1045573 |issue=1|pages=79–84|s2cid=33463483}}</ref>

=== Interior point ===

Line 246 ⟶ 282:

==== Vaidya's 89 algorithm ====

In 1989, Vaidya developed an algorithm that runs in <math>O(n^{2.5})</math> time.<ref>{{cite conference|titlechapter= Speeding-up linear programming using fast matrix multiplication | conference = 30th Annual Symposium on Foundations of Computer Science| series = FOCS |last1=Vaidya|first1=Pravin M. | title = 30th Annual Symposium on Foundations of Computer Science|year=1989| pages = 332–337| doi = 10.1109/SFCS.1989.63499 | isbn = 0-8186-1982-1}}</ref> Formally speaking, the algorithm takes <math>O( (n+d)^{1.5} n L)</math> arithmetic operations in the worst case, where <math>d</math> is the number of constraints, <math> n </math> is the number of variables, and <math>L</math> is the number of bits.

==== Input sparsity time algorithms ====

In 2015, Lee and Sidford showed that, itlinear programming can be solved in <math>\tilde O((nnz(A) + d^2)\sqrt{d}L)</math> time,<ref>{{cite conference|title= Efficient inverse maintenance and faster algorithms for linear programming | conference = FOCS '15 Foundations of Computer Science |last1=Lee|first1=Yin-Tat|last2=Sidford|first2=Aaron |year=2015| arxiv = 1503.01752 }}</ref> where <math>\tilde O</math> denotes the [[soft O notation]], and <math>nnz(A)</math> represents the number of non-zero elements, and it remains taking <math>O(n^{2.5}L)</math> in the worst case.

==== Current matrix multiplication time algorithm ====

Line 261 ⟶ 297:

{{unsolved|computer science|Does linear programming admit a strongly polynomial-time algorithm?}}

There are several open problems in the theory of linear programming, the solution of which would represent fundamental breakthroughs in mathematics and potentially major advances in our ability to solve large-scale linear programs.

* Does LP admit a [[Time complexity#Strongly and weakly polynomial time-polynomial_time|strongly polynomial]]-time algorithm?

* Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution?

* Does LP admit a polynomial-time algorithm in the real number (unit cost) model of computation?

This closely related set of problems has been cited by [[Stephen Smale]] as among the [[Smale's problems|18 greatest unsolved problems]] of the 21st century. In Smale's words, the third version of the problem "is the main unsolved problem of linear programming theory." While algorithms exist to solve linear programming in [[Strongly-polynomial_time|weakly polynomial time]], such as the [[ellipsoid method]]s and [[interior point method|interior-point techniques]], no algorithms have yet been found that allow strongly polynomial-time performance in the number of constraints and the number of variables. The development of such algorithms would be of great theoretical interest, and perhaps allow practical gains in solving large LPs as well.

Although the [[Hirsch conjecture]] was recently disproved for higher dimensions, it still leaves the following questions open.

Line 275 ⟶ 311:

The simplex algorithm and its variants fall in the family of edge-following algorithms, so named because they solve linear programming problems by moving from vertex to vertex along edges of a polytope. This means that their theoretical performance is limited by the maximum number of edges between any two vertices on the LP polytope. As a result, we are interested in knowing the maximum [[Graph diameter|graph-theoretical diameter]] of polytopal [[Graph (discrete mathematics)|graphs]]. It has been proved that all polytopes have subexponential diameter. The recent disproof of the Hirsch conjecture is the first step to prove whether any polytope has superpolynomial diameter. If any such polytopes exist, then no edge-following variant can run in polynomial time. Questions about polytope diameter are of independent mathematical interest.

Simplex pivot methods preserve primal (or dual) feasibility. On the other hand, criss-cross pivot methods do not preserve (primal or dual) feasibility{{snd}}they may visit primal feasible, dual feasible or primal-and-dual infeasible bases in any order. Pivot methods of this type have been studied since the 1970s.<ref>{{citationCite journal |last1=Anstreicher |first1=Kurt M. |last2=Terlaky |first2=Tamás needed|date=February1994 |title=A Monotonic Build-Up Simplex Algorithm for Linear Programming |journal=Operations Research |volume=42 |issue=3 |pages=556–561 |doi=10.1287/opre.42.3.556 |jstor=171894 |issn=0030-364X|doi-access=free 2019}}</ref> Essentially, these methods attempt to find the shortest pivot path on the [[arrangement polytope]] under the linear programming problem. In contrast to polytopal graphs, graphs of arrangement polytopes are known to have small diameter, allowing the possibility of strongly polynomial-time criss-cross pivot algorithm without resolving questions about the diameter of general polytopes.<ref name="FukudaTerlaky" />

== Integer unknowns ==

If all of the unknown variables are required to be integers, then the problem is called an [[integer programming]] (IP) or '''integer linear programming''' (ILP) problem. In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in many practical situations (those with bounded variables) [[NP-hard]]. '''0–1 integer programming''' or '''binary integer programming''' (BIP) is the special case of integer programming where variables are required to be 0 or 1 (rather than arbitrary integers). This problem is also classified as NP-hard, and in fact the decision version was one of [[Karp's 21 NP-complete problems]].

If only some of the unknown variables are required to be integers, then the problem is called a '''mixed integer (linear) programming''' (MIP or MILP) problem. These are generally also NP-hard because they are even more general than ILP programs.

There are however some important subclasses of IP and MIP problems that are efficiently solvable, most notably problems where the constraint matrix is [[totally unimodular]] and the right-hand sides of the constraints are integers or – more general – where the system has the [[total dual integrality]] (TDI) property.

Line 293 ⟶ 329:

== Integral linear programs ==

{{mainfurther|Integral polytope}}

A linear program in real variables is said to be '''''integral''''' if it has at least one optimal solution which is integral, i.e., made of only integer values. Likewise, a polyhedron <math>P = \{x \mid Ax \ge 0\}</math> is said to be '''''integral''''' if for all bounded feasible objective functions ''c'', the linear program <math>\{\max cx \mid x \in P\}</math> has an optimum <math>x^*</math> with integer coordinates. As observed by Edmonds and Giles in 1977, one can equivalently say that the polyhedron <math>P</math> is integral if for every bounded feasible integral objective function ''c'', the optimal ''value'' of the linear program <math>\{\max cx \mid x \in P\}</math> is an integer.

Integral linear programs are of central importance in the polyhedral aspect of [[combinatorial optimization]] since they provide an alternate characterization of a problem. Specifically, for any problem, the convex hull of the solutions is an integral polyhedron; if this polyhedron has a nice/compact description, then we can efficiently find the optimal feasible solution under any linear objective. Conversely, if we can prove that a [[linear programming relaxation]] is integral, then it is the desired description of the convex hull of feasible (integral) solutions.

Line 302 ⟶ 338:

* in an ''integral linear program,'' described in this section, variables are not constrained to be integers but rather one has proven somehow that the continuous problem always has an integral optimal value (assuming ''c'' is integral), and this optimal value may be found efficiently since all polynomial-size linear programs can be solved in polynomial time.

One common way of proving that a polyhedron is integral is to show that it is [[Totally unimodular matrix|totally unimodular]]. There are other general methods including the [[integer decomposition property]] and [[total dual integrality]]. Other specific well-known integral LPs include the matching polytope, lattice polyhedra, [[submodular]] flow]] polyhedra, and the intersection of two generalized polymatroids/''g''-polymatroids – e.g. see Schrijver 2003.

== Solvers and scripting (programming) languages ==

Line 312 ⟶ 348:

!License

!Brief info

|-

| [[Gekko (optimization software)|Gekko]]||[[MIT License]]||Open-source library for solving large-scale LP, [[Quadratic programming|QP]], [[Quadratically constrained quadratic program|QCQP]], [[Nonlinear programming|NLP]], and [[Mixed integer programming|MIP]] optimization

|-

| [[GLOP]]||[[Apache License|Apache v2]]||Google's open-source linear programming solver

|-

| [[JuMP]]||[[MPL License]]||Open-source modeling language with solvers for large-scale LP, [[Quadratic programming|QP]], [[Quadratically constrained quadratic program|QCQP]], [[Semidefinite programming|SDP]], [[Second-order cone programming|SOCP]], [[Nonlinear programming|NLP]], and [[Mixed integer programming|MIP]] optimization

|-

| [[Pyomo]]||[[BSD licenses|BSD]]||An open-source modeling language for large-scale linear, mixed integer and nonlinear optimization

|-

| [[SCIP (optimization software)|SCIP]]||[[Apache License|Apache v2]]||A general-purpose constraint integer programming solver with an emphasis on MIP. Compatible with [http://zimpl.zib.de/ Zimpl] modelling language. Free for academic use and available in source code.

| [https://www.highs.dev/ HiGHS]||[[MIT license|MIT]]||Open source serial and parallel solver for large-scale sparse linear programming (LP) models

|-

| [[SuanShu numerical library|SuanShu]]||[[Apache License|Apache v2]]||An open-source suite of optimization algorithms to solve LP, [[Quadratic programming|QP]], [[SOCP]], [[Semidefinite programming|SDP]], [[Sequential quadratic programming|SQP]] in Java

|}

Line 325 ⟶ 369:

!Brief info

|-

|[[ALGLIB]]||GPL 2+||An LP solver from ALGLIB project (C++, C#, Python)
|-
|[[Cassowary constraint solver]]||LGPL||anAn incremental constraint solving toolkit that efficiently solves systems of linear equalities and inequalities

|-

|[[COIN-OR CLP|CLP]]||CPL|| anAn LP solver from COIN-OR

|-

|[[GNU Linear Programming Kit|glpk]]||GPL|| GNU Linear Programming Kit, an LP/MILP solver with a native C [[API]] and numerous (15) third-party wrappers for other languages. Specialist support for [[flow network]]s. Bundles the [[AMPL]]-like [[GNU MathProg]] modelling language and translator.

|-

|[[lp solve]]||LGPL v2.1||An LP and [[Mixed-integer programming|MIP]] solver featuring support for the [[MPS (format)|MPS format]] and its own "lp" format, as well as custom formats through its "eXternal Language Interface" (XLI).<ref>{{cite web|url=https://web.mit.edu/lpsolve/doc/index.htm|title=lp_solve reference guide (5.5.2.5)|work=mit.edu|access-date=2023-08-10}}</ref><ref>{{cite web |url=https://lpsolve.sourceforge.net/5.5/XLI.htm |access-date=3 December 2021|title=External Language Interfaces}}</ref> Translating between model formats is also possible.<ref>{{cite web |url=http://lpsolve.sourceforge.net/5.5/lp_solve.htm |access-date=3 December 2021|title=lp_solve command}}</ref>

|[[Qoca]]||GPL||a library for incrementally solving systems of linear equations with various goal functions

|-

|[[Qoca]]||GPL||aA library for incrementally solving systems of linear equations with various goal functions

|[[R-Project]]||GPL||a programming language and software environment for statistical computing and graphics

|-

|[[R-Project]]||GPL||aA programming language and software environment for statistical computing and graphics

|}

Line 344 ⟶ 392:

!Brief info

|-

|[[AIMMS]]|| A modeling language that allows to model linear, mixed integer, and nonlinear optimization models. It also offers a tool for constraint programming. Algorithm, in the forms of heuristics or exact methods, such as Branch-and-Cut or Column Generation, can also be implemented. The tool calls an appropriate solver such as CPLEX, Gurobi or similar, to solve the optimization problem at hand. Academic licenses are free of charge.

|-

|[[ALGLIB]]|| A commercial edition of the copyleft licensed library. C++, C#, Python.

|-

|[[AMPL]]|| A popular modeling language for large-scale linear, mixed integer and nonlinear optimisation with a free student limited version available (500 variables and 500 constraints).

|-

|[[Analytica (software)|Analytica]]|| A general modeling language and interactive development environment. Its influence diagrams enable users to formulate problems as graphs with nodes for decision variables, objectives, and constraints. Analytica Optimizer Edition includes linear, mixed integer, and nonlinear solvers and selects the solver to match the problem. It also accepts other engines as plug-ins, including [[XPRESS]], Gurobi, [[Artelys Knitro]], and [[MOSEK]].

|[[APMonitor]]|| API to MATLAB and Python. Solve example [http://apmonitor.com/me575/index.php/Main/LinearProgramming Linear Programming (LP) problems] through MATLAB, Python, or a web-interface.

|-

|[[APMonitor]]|| API to MATLAB and Python. Solve example [http://apmonitor.com/me575/index.php/Main/LinearProgramming Linear Programming (LP) problems] through MATLAB, Python, or a web-interface.

|-

|[[CPLEX]]|| Popular solver with an API for several programming languages, and also has a modelling language and works with AIMMS, AMPL, [[General Algebraic Modeling System|GAMS]], MPL, OpenOpt, OPL Development Studio, and [[TOMLAB]]. Free for academic use.

Line 358 ⟶ 410:

|[[General Algebraic Modeling System|GAMS]]||

|-

|[[Gurobi Optimizer]]||

|[[Gurobi]]|| Solver with parallel algorithms for large-scale linear programs, quadratic programs and mixed-integer programs. Free for academic use.

|-

|[[IMSL Numerical Libraries]]|| Collections of math and statistical algorithms available in C/C++, Fortran, Java and C#/.NET. Optimization routines in the IMSL Libraries include unconstrained, linearly and nonlinearly constrained minimizations, and linear programming algorithms.

Line 372 ⟶ 424:

|[[Mathematica]]|| A general-purpose programming-language for mathematics, including symbolic and numerical capabilities.

|-

|[[MOSEK]]|| A solver for large scale optimization with API for several languages (C++, java, .net, Matlab and python).

|-

|[[NAG Numerical Library]]|| A collection of mathematical and statistical routines developed by the [[Numerical Algorithms Group]] for multiple programming languages (C, C++, Fortran, Visual Basic, Java and C#) and packages (MATLAB, Excel, R, LabVIEW). The Optimization chapter of the NAG Library includes routines for linear programming problems with both sparse and non-sparse linear constraint matrices, together with routines for the optimization of quadratic, nonlinear, sums of squares of linear or nonlinear functions with nonlinear, bounded or no constraints. The NAG Library has routines for both local and global optimization, and for continuous or integer problems.

|-

|[[OptimJ]]|| A Java-based modeling language for optimization with a free version available.<ref>http://www.in-ter-trans.eu/resources/Zesch_Hellingrath_2010_Integrated+Production-Distribution+Planning.pdf OptimJ used in an optimization model for mixed-model assembly lines, University of Münster</ref><ref>http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076 {{Webarchive|url=https://web.archive.org/web/20110629022829/http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076 |date=2011-06-29 }} OptimJ used in an Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games</ref>

|NMath Stats|| A general-purpose [[.NET Framework|.NET]] statistical library containing a simplex solver.<ref>{{cite web|url=http://www.centerspace.net/landing.php?id=lp|title=C# Linear Programming|work=centerspace.net}}{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

|-

|[[OptimJ]]|| A Java-based modeling language for optimization with a free version available.<ref>http://www.in-ter-trans.eu/resources/Zesch_Hellingrath_2010_Integrated+Production-Distribution+Planning.pdf OptimJ used in an optimization model for mixed-model assembly lines, University of Münster</ref><ref>http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076 OptimJ used in an Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games</ref>

|-

|[[SAS System|SAS]]/OR|| A suite of solvers for Linear, Integer, Nonlinear, Derivative-Free, Network, Combinatorial and Constraint Optimization; the [[Algebraic modeling language]] [http://support.sas.com/documentation/cdl/en/ormpug/63975/HTML/default/ormpug_optmodel_sect005.htm OPTMODEL]; and a variety of vertical solutions aimed at specific problems/markets, all of which are fully integrated with the [[SAS System]].

|-

|[[SAS System|SAS]]/OR|| A suite of solvers for Linear, Integer, Nonlinear, Derivative-Free, Network, Combinatorial and Constraint Optimization; the [[Algebraic modeling language]] [http://support.sas.com/documentation/cdl/en/ormpug/63975/HTML/default/ormpug_optmodel_sect005.htm OPTMODEL]; and a variety of vertical solutions aimed at specific problems/markets, all of which are fully integrated with the [[SAS System]].

|[[SCIP (optimization software)|SCIP]]||A general-purpose constraint integer programming solver with an emphasis on MIP. Compatible with [http://zimpl.zib.de/ Zimpl] modelling language. Free for academic use and available in source code.

|-

|[[FICO Xpress|XPRESS]]||Solver for large-scale linear programs, quadratic programs, general nonlinear and mixed-integer programs. Has API for several programming languages, also has a modelling language Mosel and works with AMPL, [[General Algebraic Modeling System|GAMS]]. Free for academic use.

Line 396 ⟶ 444:

* [[Input–output model]]

* [[Job shop scheduling]]

* [[Least absolute deviations]]

* [[Least-squares spectral analysis]]

* [[Linear algebra]]

* [[Linear production game]]

* [[Linear-fractional programming (LFP)]]

Line 401 ⟶ 452:

* [[Mathematical programming]]

* [[Nonlinear programming]]

* [[Odds algorithm]] used to solve optimal stopping problems

* [[Oriented matroid]]

* [[Quadratic programming]], a superset of linear programming

Line 415 ⟶ 467:

* {{cite journal |first=L. V. |last=Kantorovich |title=Об одном эффективном методе решения некоторых классов экстремальных проблем |trans-title=A new method of solving some classes of extremal problems |journal=[[Proceedings of the USSR Academy of Sciences|Doklady Akad Sci SSSR]] |volume=28 |year=1940 |pages=211–214 }}

* F. L. Hitchcock: ''[https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1941201224 The distribution of a product from several sources to numerous localities]'', Journal of Mathematics and Physics, 20, 1941, 224–230.

* G.B Dantzig: ''[https://books.google.com/books?hl=en&lr=&id=ZpYca36h464C&oi=fnd&pg=PA24&dq=%22Maximization+of+a+linear+function+of+variables+subject+to+linear+inequalities%22&otspg=0viWRKQVGk&sig=25NCv3tDYjTLYxCxn9deMWBn8VEPA24 Maximization of a linear function of variables subject to linear inequalities]'', 1947. Published pp.&nbsp;339–347 in T.C. Koopmans (ed.):''Activity Analysis of Production and Allocation'', New York-London 1951 (Wiley & Chapman-Hall)

* J. E. Beasley, editor. ''Advances in Linear and Integer Programming''. Oxford Science, 1996. (Collection of surveys)

* {{cite journal|pages= 103–107|jstor=3689647|doi=10.1287/moor.2.2.103|title=New Finite Pivoting Rules for the Simplex Method|journal=Mathematics of Operations Research|volume=2|issue=2|year=1977|last1=Bland|first1=Robert G.}}

* {{cite book |first=Karl-Heinz |last=Borgwardt, ''|title=The Simplex Algorithm: A Probabilistic Analysis'', |series=Algorithms and Combinatorics, Volume |volume=1, |publisher=Springer-Verlag, |date=1987.}} (Average behavior on random problems)

* Richard W. Cottle, ed. ''The Basic George B. Dantzig''. Stanford Business Books, Stanford University Press, Stanford, California, 2003. (Selected papers by [[George B. Dantzig]])

* George B. Dantzig and Mukund N. Thapa. 1997. ''Linear programming 1: Introduction''. Springer-Verlag.

* {{cite book |first1=George B. |last1=Dantzig and |first2=Mukund N. |last2=Thapa. |date=2003. ''|title=Linear Programming 2: Theory and Extensions''. |publisher=Springer-Verlag.}} (Comprehensive, covering e.g. [[simplex algorithm|pivoting]] and interior-point algorithms, large-scale problems, [[Dantzig–Wolfe decomposition|decomposition following Dantzig–Wolfe]] and [[Benders' decomposition|Benders]], and introducing [[stochastic programming]].)

* {{cite book |doi=10.1016/S0167-5060(08)70734-9|pages=185–204|chapter=A Min-Max Relation for Submodular Functions on Graphs|title=Studies in Integer Programming|volume=1|series=Annals of Discrete Mathematics|year=1977|last1=Edmonds|first1=Jack|last2=Giles|first2=Rick|isbn=978-0-7204-0765-5}}

* {{cite journal|first1=Komei|last1=Fukuda|first2=Tamás|last2=Terlaky|title=Criss-cross methods: A fresh view on pivot algorithms |journal=Mathematical Programming, Series B|volume=79|number=1–3|pages=369–395 |editor=Thomas M. Liebling |editor2=Dominique de Werra|year=1997|doi=10.1007/BF02614325|mr=1464775|citeseerx=10.1.1.36.9373|s2cid=2794181}}

Line 427 ⟶ 479:

* {{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|id=(comprehensive reference to classical approaches)}}

* Evar D. Nering and [[Albert W. Tucker]], 1993, ''Linear Programs and Related Problems'', Academic Press. (elementary<!-- but profound -->)

* {{cite book |first=M. |last=Padberg, ''|title=Linear Optimization and Extensions'', Second Edition, |publisher=Springer-Verlag, |date=1999.}} (carefully written account of primal and dual simplex algorithms and projective algorithms, with an introduction to integer linear programming – featuring the [[traveling salesman problem]] for [[Odysseus]].)

* [[{{cite book |author-link1=Christos H. Papadimitriou]] and|first1=Christos H. |last1=Papadimitriou |first2=Kenneth |last2=Steiglitz, ''|title=Combinatorial Optimization: Algorithms and Complexity'', |edition=Corrected republication with a new preface, |publisher=Dover.}} (computer science)

* {{cite journal|authorfirst=Michael J. |last=Todd |date=February 2002 | title = The many facets of linear programming | journal = Mathematical Programming | volume = 91 | issue = 3 | doi = 10.1007/s101070100261 | pages=417–436|s2cid=6464735 }} (Invited survey, from the International Symposium on Mathematical Programming.)

* {{cite book |first=Robert J. |last=Vanderbei |title=Linear Programming: Foundations and Extensions |year=2001 |publisher=Springer Verlag }}

* {{cite book | last=Vazirani | first=Vijay V. | author-link=Vijay Vazirani | title=Approximation Algorithms | year=2001 | publisher=Springer-Verlag | isbn=978-3-540-65367-7 }} (Computer science)

Line 437 ⟶ 489:

{{Library resources box |others=no}}

{{div col|colwidth=20em}}

* Dmitris Alevras and Manfred W. Padberg, ''[https://books.google.com/books?id=RAUyB8NDHJwC&printsec=frontcover#v=onepage&q&f=false Linear Optimization and Extensions: Problems and Solutions]'', Universitext, Springer-Verlag, 2001. (Problems from Padberg with solutions.)

<!-- * A. Bachem and W. Kern. ''Linear Programming Duality: An Introduction to Oriented Matroids''. Universitext. Springer-Verlag, 1992. ([[Oriented matroid|Combinatorial]]) -->

* {{cite book | author first1= Mark |last1=de Berg, |first2=Marc |last2=van Kreveld, [[|author-link3=Mark Overmars]], and|first3=Mark |last3=Overmars|first4=Otfried |last4=Schwarzkopf | year = 2000 | title = Computational Geometry | publisher = [[Springer-Verlag]] | edition = 2nd revised | isbn = 978-3-540-65620-3 | url-access = registration | url = https://archive.org/details/computationalgeo00berg }} Chapter 4: Linear Programming: pp.&nbsp;63–94. Describes a randomized half-plane intersection algorithm for linear programming.

* {{cite book|author = [[Michael R. Garey]] and [[David S. Johnson]] | year = 1979 | title = Computers and Intractability: A Guide to the Theory of NP-Completeness | publisher = W.H. Freeman | isbn = 978-0-7167-1045-5| title-link = Computers and Intractability: A Guide to the Theory of NP-Completeness }} A6: MP1: INTEGER PROGRAMMING, pg.245. (computer science, complexity theory)

* {{Cite Gartner Matousek 2006}} (elementary introduction for mathematicians and computer scientists)

Line 445 ⟶ 497:

* {{cite book|author = Alexander Schrijver | year = 2003 | title = Combinatorial optimization: polyhedra and efficiency | publisher = Springer}}

* Alexander Schrijver, ''Theory of Linear and Integer Programming''. John Wiley & sons, 1998, {{isbn|0-471-98232-6}} (mathematical)

* {{cite book|author1=Gerard Sierksma|author2=Yori Zwols|title=Linear and Integer Optimization: Theory and Practice|year=2015|publisher=CRC Press|isbn=978-1-498-71016-9}}; with online solver: https://online-optimizer.appspot.com/

* {{cite book|author1=Gerard Sierksma|author2=Diptesh Ghosh|title=Networks in Action; Text and Computer Exercises in Network Optimization|year=2010|publisher=Springer|isbn=978-1-4419-5512-8}} (linear optimization modeling)

* H. P. Williams, ''[https://books.google.com/books?id=YJRh0tOes7UC&printsec=frontcover#v=onepage&q&f=false Model Building in Mathematical Programming]'', Fifth Edition, 2013. (Modeling)

* Stephen J. Wright, 1997, ''[https://books.google.com/books?id=oQdBzXhZeUkC&printsec=frontcover#v=onepage&q&f=false Primal-Dual Interior-Point Methods]'', SIAM. (Graduate level)

* [[Yinyu Ye]], 1997, ''Interior Point Algorithms: Theory and Analysis'', Wiley. (Advanced graduate-level)

* [[Günter M. Ziegler|Ziegler, Günter M.]], Chapters 1–3 and 6–7 in ''Lectures on Polytopes'', Springer-Verlag, New York, 1994. (Geometry)

Line 454 ⟶ 506:

==External links==

{{Commons category}}

{{Commonscat}}

*[http://people.brunel.ac.uk/~mastjjb/jeb/or/lp.html Guidance On Formulating LP Problems]

*[http://glossary.computing.society.informs.org/ Mathematical Programming Glossary]

*[httphttps://lpsolve.sourceforge.net/4.0/LinearProgrammingFAQ.htm The Linear Programming FAQ]

*[http://plato.asu.edu/bench.html Benchmarks For Optimisation Software]