NationMaster - Encyclopedia: Linear programming

In mathematics, linear programming (LP) problems involve the optimization of a linear objective function, subject to linear equality and inequality constraints. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ... For other uses, see Linear (disambiguation). ... Optimization is a branch of mathematics which is concerned with finding maxima and minima of real-valued functions. ... Constraint is an equation that defines a restriction of solutions of an optimization problem to a so called feasible set. ...

Put very informally, LP problems determine the way to achieve the best outcome (such as maximum profit or lowest cost) given some list of requirements represented as linear equations.

More formally, given a polytope (for example, a polygon or a polyhedron), and a real-valued affine function In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions. ... Look up polygon in Wiktionary, the free dictionary. ... For the game magazine, see Polyhedron (magazine). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The word linear comes from the Latin word linearis, which means created by lines. ...

$f(x_1, x_2, dots, x_n)=a_1x_1+a_2x_2+cdots +a_nx_n+b,$

defined on this polytope, the goal is to find a point in the polytope where this function has the smallest (or largest) value. Such points may not exist, but if they do, searching through the polytope vertices is guaranteed to find at least one of them.

Linear programs are problems that can be expressed in canonical form:

Maximize $mathbf{c}^T mathbf{x}$

Subject to $Amathbf{x} leq mathbf{b}$

$mathbf{x}$ represents the vector of variables (to be determined), while $mathbf{c}$ and $mathbf{b}$ are vectors of (known) coefficients and $mathbf{A}$ is a (known) matrix of coefficients. The expression to be maximized or minimized is called the objective function ( $mathbf{c}^T mathbf{x}$ in this case). The equations $Amathbf{x} leq mathbf{b}$ are the constraints which specify a convex polyhedron over which the objective function is to be optimized.

Linear programming can be applied to various fields of study. Most extensively it is used in business and economic situations, but can also be utilized for some engineering problems. Some industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.

1 History of linear programming
2 Uses
3 Standard form
- 3.1 Example
4 Augmented form (slack form)
- 4.1 Example
5 Duality
- 5.1 Example
6 Complementary slackness
7 Theory
8 Algorithms
9 Open problems and recent work
10 Integer unknowns
11 Solvers and Scripting (Programming) Languages
12 See also
13 References
14 Further reading
15 External links

History of linear programming

The problem of solving a system of linear inequalities dates back at least as far as Fourier, after whom the method of Fourier-Motzkin elimination is named. Linear programming arose as a mathematical model developed during the second world war to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Postwar, many industries found its use in their daily planning. Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... Fourierâ€“Motzkin elimination is a mathematical algorithm for solving a system of linear inequalities. ... Combatants Allied powers: China France Great Britain Soviet Union United States and others Axis powers: Germany Italy Japan and others Commanders Chiang Kai-shek Charles de Gaulle Winston Churchill Joseph Stalin Franklin Roosevelt Adolf Hitler Benito Mussolini Hideki TÅjÅ Casualties Military dead: 17,000,000 Civilian dead: 33,000...

The founders of the subject are George B. Dantzig, who published the simplex method in 1947, John von Neumann, who developed the theory of the duality in the same year, and Leonid Kantorovich, a Russian mathematician who used similar techniques in economics before Dantzig and won the Nobel prize in 1975 in economics. The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger major theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior point method for solving linear programming problems. George Bernard Dantzig (8 November 1914 â€“ 13 May 2005) was an American mathematician who introduced the simplex algorithm and independently discovered linear programming some years after it was initially invented by Soviet economist and mathematician Leonid Kantorovich. ... In mathematical optimization theory, the simplex algorithm, created by the American mathematician George Dantzig in 1947, is a popular technique for numerical solution of the linear programming problem. ... For other persons named John Neumann, see John Neumann (disambiguation). ... Leonid V. Kantorovich. ... Leonid Khachiyan Leonid Khachiyan (May 3, 1952 - April 29, 2005) was a Russian-born mathematician who taught Computer Science at Rutgers University. ... Narendra K. Karmarkar (b. ... Interior point methods are a certain class of algorithms to solve linear and nonlinear convex optimization problems. ...

Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the 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 optimal solutions that must be checked.

Uses

Linear programming is an important field of optimization for several reasons. Many practical problems in operations research can be expressed as linear programming problems. Certain special cases of linear programming, such as network flow problems and 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 LP 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 is heavily used in microeconomics and business management, either to maximize the income or minimize the costs of a production scheme. Some examples are food blending, inventory management, portfolio and finance management, resource allocation for human and machine resources, planning advertisement campaigns, etc. Operations Research or Operational Research (OR) is an interdisciplinary branch of mathematics which uses methods like mathematical modeling, statistics, and algorithms to arrive at optimal or good decisions in complex problems which are concerned with optimizing the maxima (profit, faster assembly line, greater crop yield, higher bandwidth, etc) or minima... Microeconomics (or price theory) is a branch of economics that studies how individuals, households, and firms make decisions to allocate limited resources,[1] typically in markets where goods or services are being bought and sold. ...

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 function to be maximized

e.g. maximize $c_1 x_1 + c_2 x_2,$

Problem constraints of the following form

e.g. $a_{11} x_1 + a_{12} x_2 le b_1$

$a_{21} x_1 + a_{22} x_2 le b_2$

$a_{31} x_1 + a_{32} x_2 le b_3$

Non-negative variables

e.g. $x_1 ge 0$

$x_2 ge 0$

The problem is usually expressed in matrix form, and then becomes: In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

maximize $mathbf{c}^T mathbf{x}$

subject to $mathbf{A}mathbf{x} le mathbf{b}, , mathbf{x} ge 0$

Other forms, such as minimization problems, problems with constraints on alternative forms, as well as problems involving negative variables can always be rewritten into an equivalent problem in standard form. In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ...

Example

Suppose that a farmer has a piece of farm land, say A square kilometres large, to be planted with either wheat or barley or some combination of the two. The farmer has a limited permissible amount F of fertilizer and P of insecticide which can be used, each of which is required in different amounts per unit area for wheat (F₁, P₁) and barley (F₂, P₂). Let S₁ be the selling price of wheat, and S₂ the price of barley. If we denote the area planted with wheat and barley by x₁ and x₂ respectively, then the optimal number of square kilometres to plant with wheat vs barley can be expressed as a linear programming problem:

maximize $S_1 x_1 + S_2 x_2 ,$		(maximize the revenue — revenue is the "objective function")
subject to	$x_1 + x_2 le A$	(limit on total area)
	$F_1 x_1 + F_2 x_2 le F$	(limit on fertilizer)
	$P_1 x_1 + P_2 x_2 le P$	(limit on insecticide)
	$x_1 ge 0,, x_2 ge 0$	(cannot plant a negative area)

Which in matrix form becomes:

maximize $begin{bmatrix} S_1 & S_2 end{bmatrix} begin{bmatrix} x_1 x_2 end{bmatrix}$

subject to $begin{bmatrix} 1 & 1 F_1 & F_2 P_1 & P_2 end{bmatrix} begin{bmatrix} x_1 x_2 end{bmatrix} le begin{bmatrix} A F P end{bmatrix}, , begin{bmatrix} x_1 x_2 end{bmatrix} ge 0$

Augmented form (slack form)

Linear programming problems must be converted into augmented form before being solved by the simplex algorithm. This form introduces non-negative slack variables to replace inequalities with equalities in the constraints. The problem can then be written in the following form: In mathematical optimization theory, the simplex algorithm, created by the American mathematician George Dantzig in 1947, is a popular technique for numerical solution of the linear programming problem. ... In Linear programming a slack variable is a variable which is added to a constraint to turn the inequality into an equation. ...

Maximize Z in:

$begin{bmatrix} 1 & -mathbf{c}^T & 0 0 & mathbf{A} & mathbf{I} end{bmatrix} begin{bmatrix} Z mathbf{x} mathbf{x}_s end{bmatrix} = begin{bmatrix} 0 mathbf{b} end{bmatrix}$

$mathbf{x}, , mathbf{x}_s ge 0$

where $mathbf{x}_s$ are the newly introduced slack variables, and Z is the variable to be maximized.

Example

The example above becomes as follows when converted into augmented form:

maximize $S_1 x_1 + S_2 x_2,$		(objective function)
subject to	$x_1 + x_2 + x_3 = A,$	(augmented constraint)
	$F_1 x_1 + F_2 x_2 + x_4 = F,$	(augmented constraint)
	$P_1 x_1 + P_2 x_2 + x_5 = P,$	(augmented constraint)
	$x_1,x_2,x_3,x_4,x_5 ge 0$

where $x_3,x_4,x_5,$ are (non-negative) slack variables.

Which in matrix form becomes:

Maximize Z in:

$begin{bmatrix} 1 & -S_1 & -S_2 & 0 & 0 & 0 0 & 1 & 1 & 1 & 0 & 0 0 & F_1 & F_2 & 0 & 1 & 0 0 & P_1 & P_2 & 0 & 0 & 1 end{bmatrix} begin{bmatrix} Z x_1 x_2 x_3 x_4 x_5 end{bmatrix} = begin{bmatrix} 0 A F P end{bmatrix}, , begin{bmatrix} x_1 x_2 x_3 x_4 x_5 end{bmatrix} ge 0$

Duality

Every linear programming problem, referred to as a primal problem, can be converted into a dual problem, which provides an upper bound to the optimal value of the primal problem. In matrix form, we can express the primal problem as: This page may meet Wikipedias criteria for speedy deletion. ...

maximize $mathbf{c}^T mathbf{x}$

subject to $mathbf{A}mathbf{x} le mathbf{b}, , mathbf{x} ge 0$

The corresponding dual problem is:

minimize $mathbf{b}^T mathbf{y}$

subject to $mathbf{A}^T mathbf{y} ge mathbf{c}, , mathbf{y} ge 0$

where y is used instead of x as variable vector.

There are two ideas fundamental to duality theory. One is the fact that the dual of a dual linear program is the original primal linear program. Additionally, every feasible solution for a linear program gives a bound on the optimal value of the objective function of its dual. The weak duality theorem states that the objective function value of the dual at any feasible solution is always greater than or equal to the objective function value of the primal at any feasible solution. The strong duality theorem states that if the primal has an optimal solution, x*, then the dual also has an optimal solution, y*, such that c^Tx*=b^Ty*.

A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible.

Example

Following the above example of the farmer with some A land, F fertilizer and P insecticide, the farmer can tell others that he has no way to earn more than a specific amount of profit with the following scheme: to claim that with his available method of earning, each kilometre of land can give him no more than $y A$ , each amount of fertilizer can earn him no more than $y F$ , and each amount of insecticide can earn him no more than $y P$ . Then he can tell others that the most he can earn is $A y A + F y F + P y P$ . In order to find the best (lowest) claim he can make, he can set $y A$ , $y F$ and $y P$ by using the following linear programming problem:

minimize $A y_A + F y_F + P y_P ,$		(minimize the revenue bound — revenue bound is the "objective function")
subject to	$y_A + F_1 y_F + P_1 y_P ge S_1$	(he can earn no more by growing wheat)
	$y_A + F_2 y_F + P_2 y_P ge S_2$	(he can earn no more by growing barley)
	$y_A ge 0,, y_F ge 0,, y_P ge 0$	(cannot claim negative revenue on resource)

Which in matrix form becomes:

minimize $begin{bmatrix} A & F & P end{bmatrix} begin{bmatrix} y_A y_F y_P end{bmatrix}$

subject to $begin{bmatrix} 1 & F_1 & P_1 1 & F_2 & P_2 end{bmatrix} begin{bmatrix} y_A y_F y_P end{bmatrix} ge begin{bmatrix} S_1 S_2 end{bmatrix}, , begin{bmatrix} y_A y_F y_P end{bmatrix} ge 0$

Note that each variable in the primal problem (amount of wheat/barley to grow) correspond to an inequality in the dual problem (revenue obtained by wheat/barley), and each variable in the dual problem (revenue bound provided by each resource) correspond to an inequality in the primal problem (limit on each resource).

Since each inequality can be replaced by an equality and a slack variable, this means each primal variable correspond to a dual slack variable, and each dual variable correspond to a primal slack variable. This relation allows us to complementary slackness.

Complementary slackness

It is possible to obtain an optimal solution to the dual when only an optimal solution to the primal is known using the complementary slackness theorem. The theorem states:

Suppose that x = (x₁, x₂, . . ., x_n) is primal feasible and that y = (y₁, y₂, . . . , y_m) is dual feasible. Let (w₁, w₂, . . ., w_m) denote the corresponding primal slack variables, and let (z₁, z₂, . . . , z_n) denote the corresponding dual slack variables. Then x and y are optimal for their respective problems if and only if x_jz_j = 0, for j = 1, 2, . . . , n, w_iy_i = 0, for i = 1, 2, . . . , m.

So if the ith slack variable of the primal is not zero, then the ith variable of the dual is equal zero. Likewise, if the jth slack variable of the dual is not zero, then the jth variable of the primal is equal to zero.

Theory

Geometrically, the linear constraints define a convex polyhedron, which is called the feasible region. Since the objective function is also linear, hence a convex function, all local optima are automatically global optima (by the KKT theorem). The linearity of the objective function also implies that the set of optimal solutions is the convex hull of a finite set of points - usually a single point. Look up Convex set in Wiktionary, the free dictionary. ... For the game magazine, see Polyhedron (magazine). ... In optimization (a branch of mathematics), a candidate solution is a member of a set of possible solutions to a given problem. ... In mathematics, the Karush-Kuhn-Tucker conditions (also known as the Kuhn-Tucker or the KKT conditions) are necessary for a solution in nonlinear programming to be optimal. ... Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. // For planar objects, i. ...

There are two situations in which no optimal solution can be found. First, if the constraints contradict each other (for instance, x ≥ 2 and x ≤ 1) then the feasible region is empty and there can be no optimal solution, since there are no solutions at all. In this case, the LP is said to be infeasible.

Alternatively, the polyhedron can be unbounded in the direction of the objective function (for example: maximize x₁ + 3 x₂ subject to x₁ ≥ 0, x₂ ≥ 0, x₁ + x₂ ≥ 10), in which case there is no optimal solution since solutions with arbitrarily high values of the objective function can be constructed. For the game magazine, see Polyhedron (magazine). ...

Barring these two pathological conditions (which are often ruled out by resource constraints integral to the problem being represented, as above), the optimum is always attained at a vertex of the polyhedron. However, the optimum is not necessarily unique: it is possible to have a set of optimal solutions covering an edge or face of the polyhedron, or even the entire polyhedron (This last situation would occur if the objective function were constant).

Algorithms

A series of linear constraints on two variables produces a region of possible values for those variables. Solvable problems will have a feasible region in the shape of a simple polygon.

The simplex algorithm, developed by George Dantzig, solves LP problems by constructing an admissible solution at a vertex of the polyhedron and then walking along edges of the polyhedron to vertices with successively higher values of the objective function until the optimum is reached. Although this algorithm is quite efficient in practice and can be guaranteed to find the global optimum if certain precautions against cycling are taken, it has poor worst-case behavior: it is possible to construct a linear programming problem for which the simplex method takes a number of steps exponential in the problem size. In fact, for some time it was not known whether the linear programming problem was solvable in polynomial time (complexity class P). A graph showing how a series of linear constraints on two variables produce a feasible region in a linear programming problem. ... A graph showing how a series of linear constraints on two variables produce a feasible region in a linear programming problem. ... A simple concave hexagon In geometry, two edges of a polygon may cross or even overlap in general. ... In mathematical optimization theory, the simplex algorithm, created by the American mathematician George Dantzig in 1947, is a popular technique for numerical solution of the linear programming problem. ... George Bernard Dantzig (8 November 1914 â€“ 13 May 2005) was an American mathematician who introduced the simplex algorithm and independently discovered linear programming some years after it was initially invented by Soviet economist and mathematician Leonid Kantorovich. ... Flowcharts are often used to graphically represent algorithms. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ...

This long standing issue was resolved by Leonid Khachiyan in 1979 with the introduction of the ellipsoid method, the first worst-case polynomial-time algorithm for linear programming. To solve a problem which has n variables and can be encoded in L input bits, this algorithm uses O(n⁴L) arithmetic operations on numbers with O(L) digits. It consists of a specialization of the nonlinear optimization technique developed by Naum Z. Shor, generalizing the ellipsoid method for convex optimization proposed by Arkadi Nemirovski, a 2003 John von Neumann Theory Prize winner, and D. Yudin. Leonid Khachiyan Leonid Khachiyan (May 3, 1952 - April 29, 2005) was a Russian-born mathematician who taught Computer Science at Rutgers University. ... Also: 1979 by Smashing Pumpkins. ... The ellipsoid method is an algorithm for solving linear programs. ... In mathematics, nonlinear programming (NLP) is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function is nonlinear. ... Convex optimization is a subfield of mathematical optimization. ... Year 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ... The John von Neumann Theory Prize of the Institute for Operations Research and Management Science (INFORMS, previously The Institute of Management Science, TIMS, and the Operations Research Society of America, ORSA) is awarded annually to an individual (or sometimes group) who have made fundamental and sustained contributions to theory in...

Khachiyan's algorithm was of landmark importance for establishing the polynomial-time solvability of linear programs. The algorithm had little practical impact, as the simplex method is more efficient for all but specially constructed families of linear programs. However, it inspired new lines of research in linear programming with the development of interior point methods, which can be implemented as a practical tool. In contrast to the simplex algorithm, which finds the optimal solution by progresses along points on the boundary of a polyhedral set, interior point methods move through the interior of the feasible region. Interior point methods are a certain class of algorithms to solve linear and nonlinear convex optimization problems. ...

In 1984, N. Karmarkar proposed a new interior point projective method for linear programming. Karmarkar's algorithm not only improved on Khachiyan's theoretical worst-case polynomial bound (giving $O (n 3.5 L)$ ), but also promised dramatic practical performance improvements over the simplex method. Since then, many interior point methods have been proposed and analyzed. Early successful implementations were based on affine scaling variants of the method. For both theoretical and practical properties, barrier function or path-following methods are the most common recently. This article is about the year. ... Narendra K. Karmarkar (b. ... In mathematics, Karmarkars algorithm is an algorithm for solving linear programming problems. ... In mathematics, Karmarkars algorithm is an algorithm for solving linear programming problems. ... In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible region (Nocedal and Wright 1999). ...

The current opinion is that the efficiency of good implementations of simplex-based methods and interior point methods is similar for routine applications of linear programming.

LP solvers are in widespread use for optimization of various problems in industry, such as optimization of flow in transportation networks, many of which can be transformed into linear programming problems only with some difficulty.

Open problems and recent work

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 strongly polynomial-time algorithm?
Does LP admit a strongly polynomial algorithm to find a strictly complementary solution?
Does LP admit a polynomial 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 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 weakly polynomial time, such as the ellipsoid methods and 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. Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. ... Smales problems refers to a list of eighteen unsolved problems in mathematics, proposed by Steve Smale in 2000. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the run time, m(n), is no greater than a polynomial function of the problem size, n. ... The ellipsoid method is an algorithm for solving linear programs. ... Interior point methods are a certain class of algorithms to solve linear and nonlinear convex optimization problems. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the run time, m(n), is no greater than a polynomial function of the problem size, n. ...

Are there pivot rules which lead to polynomial-time Simplex variants?
Do all polyhedral graphs have polynomially-bounded diameter?
Is the Hirsch conjecture true for polyhedral graphs?

These questions relate to the performance analysis and development of Simplex-like methods. The immense efficiency of the Simplex algorithm in practice despite its exponential-time theoretical performance hints that there may be variations of Simplex that run in polynomial or even strongly polynomial time. It would be of great practical and theoretical significance to know whether any such variants exist, particularly as an approach to deciding if LP can be solved in strongly polynomial time. In mathematical programming and geometry, Hirschs conjecture states that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n âˆ’ d. ...

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 polyhedron. This means that their theoretical performance is limited by the maximum number of edges between any two vertices on the LP polyhedron. As a result, we are interested in knowing the maximum graph-theoretical diameter of polyhedral graphs. It has been proved that all polyhedra have subexponential diameter, and all experimentally observed polyhedra have linear diameter, it is presently unknown whether any polyhedron has superpolynomial or even superlinear diameter. If any such polyhedra exist, then no edge-following variant can run in polynomial or linear time, respectively. Questions about polyhedron diameter are of independent mathematical interest. In the mathematical subfield of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path connecting them. ...

Simplex pivot methods preserve primal (or dual) feasibility. On the other hand, criss-cross pivot methods do not preserve (primal or dual) feasibility --- 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. Essentially, these methods attempt to find the shortest pivot path on the arrangement polytope under the linear programming problem. In contrast to polyhedral graphs, graphs of arrangement polytopes have small diameter, allowing the possibility of strongly polynomial-time criss-cross pivot method without resolving questions about the diameter of general polyhedra.

Integer unknowns

If the unknown variables are all 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. In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing... In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing... One of the most important results of computational complexity theory was Stephen Cooks 1971 paper that demonstrated the first NP-complete problem, the boolean satisfiability problem. ...

If only some of the unknown variables are required to be integers, then the problem is called a mixed integer programming (MIP) problem. These are generally also NP-hard. In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing...

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. In mathematics, a unimodular matrix is a square matrix with determinant +1 or -1. ...

Advanced algorithms for solving integer linear programs include:

cutting-plane method
branch and bound
branch and cut
branch and price
if the problem has some extra structure, it may be possible to apply delayed column generation.

In mathematics, more specifically in optimization, the cutting-plane method is a procedure used to find integer solutions of a linear program. ... Branch and bound is a general algorithmic method for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Delayed column generation is an efficient algorithm for solving larger integer linear programs. ...

Solvers and Scripting (Programming) Languages

AIMMS is an advanced development environment for building optimization based decision support applications and advanced planning systems. ... Cassowary is an incremental constraint solving toolkit that efficiently solves systems of linear equalities and inequalities. ... SYMPHONY is an acronym standing for Single- or Multi-Process Optimization over Networks. ... CPLEX is an optimization software package. ... The General Algebraic Modeling System (GAMS) is a high-level modeling system for mathematical programming and optimization. ... The GNU Linear Programming Kit (GLPK) is a software package intended for solving large-scale linear programming (LP), mixed integer programming (MIP), and other related problems. ... Lingo is an American television game show that GSN produced along with other companies. ... For other uses, see Mathematica (disambiguation). ... MINTO is a integer programming solver which uses branch and bound algorithm. ... Qoca is a GPL library for incrementally solving systems of linear equations with various goal functions. ... The SAS System (originally Statistical Analysis System) is an integrated system of software products provided by SAS Institute that enables the programmer to perform: data entry, retrieval, management, and mining report writing and graphics statistical and mathematical analysis business planning, forecasting, and decision support operations research and project management quality...

References

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External links

Software

AIMMS Optimization Modeling AIMMS — include linear programming in industry solutions (free trial license available);
Calc Optimization Solver — Kohei Yoshida’s spreadsheet add-in for OpenOffice.org Calc
CGAL — The Computational Geometry Algorithms Library includes a linear solver, which is exact and optimized for problems with few constraints or few variables
COIN-OR — COmputational INfrastructure for Operations Research, open-source library
GAMS — General Algebraic Modeling System
Cplex — Commercial library for linear programming
GLPK — GNU Linear Programming Kit; open source LP software
HOPDM — Higher Order Primal Dual Method
LINDO — LP, IP, Global solver/modeling language
lp_solve — open-source solver with C library
MOSEK — Optimization software for LP,IP,QP,SOCP and MIP. Free trial is available. Free for students.
Mathematica — General technical computing system includes large scale linear programming support
OptimJ — Java-based algebraic modeling language; free evaluation version.
Premium Solver — Spreadsheet add-in
SAS — Includes optimization modeling language and solvers for LP, MILP, QP, and NLP
What's Best! — Spreadsheet add-in
Xpress-MP — Optimization software free to students
Vanguard Linear Programming Optimization Software
QSopt Optimization software for LP (free for research purposes).
Microarray Data Classification Server (MDCS) based on linear programming
Linear programming and linear goal programming A freeware program for MS-DOS
Simplex Method Tool A quick-loading web page
IBM's article on GLPK A technical article on GLPK with an introduction to Linear Programming by IBM
Orstat2000 — Includes easy-to-use modules for linear and integer programming (free for educational purposes).
Trilinos is a set of scientific tools with a few linear and non-linear program solvers.
TOMLAB provides optimization solvers in MATLAB, LabVIEW and .NET.
SCIP It can be used as a standalone program to solve mixed integer programs given in MPS Format.
GIPALS32.DLL — Linear programming callable library for Windows (free trial and academic license available).
CVXOPT — Python library that can be used for linear, second-order cone and semidefinite programming (open source, GPLv3)

Categories: Optimization | Operations research

Hidden categories: Articles lacking sources from June 2008 | All articles lacking sources

AIMMS is an advanced development environment for building optimization based decision support applications and advanced planning systems. ... The GNU Linear Programming Kit (GLPK) is a software package intended for solving large-scale linear programming (LP), mixed integer programming (MIP), and other related problems. ...

Results from FactBites:

Linear Programming - Wolfram Mathematica (1834 words)
	Linear programming problems are optimization problems where the objective function and constraints are all linear.
	The simplex and revised simplex algorithms solve a linear programming problem by moving along the edges of the polytope defined by the constraints, from vertices to vertices with successively smaller values of the objective function, until the minimum is reached.
	The simplex and revised simplex algorithms solve linear programming problems by constructing a feasible solution at a vertex of the polytope defined by the constraints, and then moving along the edges of the polytope to vertices with successively smaller values of the objective function until the minimum is reached.

Linear programming - Wikipedia, the free encyclopedia (1542 words)
	In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear.
	Linear programming is an important field of optimization for several reasons.
	Leonid Kantorovich was one of founders of linear programming

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Contents

History of linear programming GA_googleFillSlot("encyclopedia_square");

Uses

Standard form

Example

Augmented form (slack form)

Example

Duality

Example

Complementary slackness

Theory

Algorithms

Open problems and recent work

Integer unknowns

Solvers and Scripting (Programming) Languages

See also

References

Further reading

External links

History of linear programming