Linear programming
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 relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization).
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a realvalued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the largest (or smallest) value if such a point exists.
Linear programs are problems that can be expressed in standard form as
 [math]\displaystyle{ \begin{align} & \text{Find a vector} && \mathbf{x} \\ & \text{that maximizes} && \mathbf{c}^\mathsf{T} \mathbf{x}\\ & \text{subject to} && A \mathbf{x} \le \mathbf{b} \\ & \text{and} && \mathbf{x} \ge \mathbf{0}. \end{align} }[/math]
Here the components of [math]\displaystyle{ \mathbf{x} }[/math] are the variables to be determined, [math]\displaystyle{ \mathbf{c} }[/math] and [math]\displaystyle{ \mathbf{b} }[/math] are given vectors, and [math]\displaystyle{ A }[/math] is a given matrix. The function whose value is to be maximized ([math]\displaystyle{ \mathbf x\mapsto\mathbf{c}^\mathsf{T}\mathbf{x} }[/math] in this case) is called the objective function. The constraints [math]\displaystyle{ A \mathbf{x} \le \mathbf{b} }[/math] and [math]\displaystyle{ \mathbf{x} \geq \mathbf{0} }[/math] specify a convex polytope over which the objective function is to be optimized.
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 some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proven useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.
History
The problem of solving a system of linear inequalities dates back at least as far as Fourier, who in 1827 published a method for solving them,^{[1]} 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.
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.
Despite its initial obscurity, the wartime successes propelled linear programming into the spotlight. PostWWII, 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 decisionmaking processes.^{[2]}
Kantorovich's work was initially neglected in the USSR.^{[3]} About the same time as Kantorovich, the DutchAmerican economist T. C. Koopmans formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 Nobel Memorial Prize in Economic Sciences.^{[1]} In 1941, Frank Lauren Hitchcock also formulated transportation problems as linear programs and gave a solution very similar to the later simplex method.^{[4]} Hitchcock had died in 1957, and the Nobel Memorial Prize is not awarded posthumously.
From 1946 to 1947 George B. Dantzig independently developed general linear programming formulation to use for planning problems in the US Air Force.^{[5]} In 1947, Dantzig also invented the simplex method that, for the first time efficiently, tackled the linear programming problem in most cases.^{[5]} When Dantzig arranged a meeting with John von Neumann to discuss his simplex method, Neumann immediately conjectured the theory of duality by realizing that the problem he had been working in game theory was equivalent.^{[5]} Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948.^{[3]} Dantzig's work was made available to public in 1951. In the postwar 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 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.
The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979,^{[6]} but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interiorpoint method for solving linearprogramming problems.^{[7]}
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.^{[3]} Certain special cases of linear programming, such as network flow problems and multicommodity flow problems, are considered important enough to have much research on specialized algorithms. A number of algorithms for other types of optimization problems work by solving linear programming problems as subproblems. 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, and technology. Although the modern management issues are everchanging, most companies would like to maximize profits and minimize costs with limited resources. Google also uses linear programming to stabilize YouTube videos.^{[8]}
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]\displaystyle{ f(x_{1},x_{2}) = c_1 x_1 + c_2 x_2 }[/math]
 Problem constraints of the following form
 e.g.
 [math]\displaystyle{ \begin{matrix} a_{11} x_1 + a_{12} x_2 &\leq b_1 \\ a_{21} x_1 + a_{22} x_2 &\leq b_2 \\ a_{31} x_1 + a_{32} x_2 &\leq b_3 \\ \end{matrix} }[/math]
 Nonnegative variables
 e.g.
 [math]\displaystyle{ \begin{matrix} x_1 \geq 0 \\ x_2 \geq 0 \end{matrix} }[/math]
The problem is usually expressed in matrix form, and then becomes:
 [math]\displaystyle{ \max \{\, \mathbf{c}^\mathsf{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, and problems involving negative variables can always be rewritten into an equivalent problem in standard form.
Example
Suppose that a farmer has a piece of farm land, say L km^{2}, to be planted with either wheat or barley or some combination of the two. The farmer has a limited amount of fertilizer, F kilograms, and pesticide, P kilograms. Every square kilometer of wheat requires F_{1} kilograms of fertilizer and P_{1} kilograms of pesticide, while every square kilometer of barley requires F_{2} kilograms of fertilizer and P_{2} kilograms of pesticide. Let S_{1} be the selling price of wheat per square kilometer, and S_{2} be the selling price of barley. 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:
Maximize: [math]\displaystyle{ 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")  
Subject to:  [math]\displaystyle{ x_1 + x_2\leq L }[/math]  (limit on total area) 
[math]\displaystyle{ F_1\cdot x_1+F_2\cdot x_2\leq F }[/math]  (limit on fertilizer)  
[math]\displaystyle{ P_1\cdot x_1 + P_2\cdot x_2\leq P }[/math]  (limit on pesticide)  
[math]\displaystyle{ x_1\geq 0, x_2\geq 0 }[/math]  (cannot plant a negative area). 
In matrix form this becomes:
 maximize [math]\displaystyle{ \begin{bmatrix} S_1 & S_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} }[/math]
 subject to [math]\displaystyle{ \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} L \\ F \\ P \end{bmatrix}, \, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \ge \begin{bmatrix} 0 \\ 0 \end{bmatrix}. }[/math]
Augmented form (slack form)
Linear programming problems can be converted into an augmented form in order to apply the common form of the simplex algorithm. This form introduces nonnegative slack variables to replace inequalities with equalities in the constraints. The problems can then be written in the following block matrix form:
 Maximize [math]\displaystyle{ z }[/math]:
 [math]\displaystyle{ \begin{bmatrix} 1 & \mathbf{c}^\mathsf{T} & 0 \\ 0 & \mathbf{A} & \mathbf{I} \end{bmatrix} \begin{bmatrix} z \\ \mathbf{x} \\ \mathbf{s} \end{bmatrix} = \begin{bmatrix} 0 \\ \mathbf{b} \end{bmatrix} }[/math]
 [math]\displaystyle{ \mathbf{x} \ge 0, \mathbf{s} \ge 0 }[/math]
where [math]\displaystyle{ \mathbf{s} }[/math] are the newly introduced slack variables, [math]\displaystyle{ \mathbf{x} }[/math] are the decision variables, and [math]\displaystyle{ z }[/math] is the variable to be maximized.
Example
The example above is converted into the following augmented form:
Maximize: [math]\displaystyle{ S_1\cdot x_1+S_2\cdot x_2 }[/math] (objective function) subject to: [math]\displaystyle{ x_1 + x_2 + x_3 = L }[/math] (augmented constraint) [math]\displaystyle{ F_1\cdot x_1+F_2\cdot x_2 + x_4 = F }[/math] (augmented constraint) [math]\displaystyle{ P_1\cdot x_1 + P_2\cdot x_2 + x_5 = P }[/math] (augmented constraint) [math]\displaystyle{ x_1,x_2,x_3,x_4,x_5 \ge 0. }[/math]
where [math]\displaystyle{ x_3, x_4, x_5 }[/math] are (nonnegative) slack variables, representing in this example the unused area, the amount of unused fertilizer, and the amount of unused pesticide.
In matrix form this becomes:
 Maximize [math]\displaystyle{ z }[/math]:
 [math]\displaystyle{ \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 \\ L \\ F \\ P \end{bmatrix}, \, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix} \ge 0. }[/math]
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:
 Maximize c^{T}x subject to Ax ≤ b, x ≥ 0;
 with the corresponding symmetric dual problem,
 Minimize b^{T}y subject to A^{T}y ≥ c, y ≥ 0.
An alternative primal formulation is:
 Maximize c^{T}x subject to Ax ≤ b;
 with the corresponding asymmetric dual problem,
 Minimize b^{T}y subject to A^{T}y = c, y ≥ 0.
There are two ideas fundamental to duality theory. One is the fact that (for the symmetric dual) 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^{*}, and c^{T}x^{*}=b^{T}y^{*}.
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. See dual linear program for details and several more examples.
Variations
Covering/packing dualities
Covering/packingproblem pairs  



A covering LP is a linear program of the form:
 Minimize: b^{T}y,
 subject to: A^{T}y ≥ c, y ≥ 0,
such that the matrix A and the vectors b and c are nonnegative.
The dual of a covering LP is a packing LP, a linear program of the form:
 Maximize: c^{T}x,
 subject to: Ax ≤ b, x ≥ 0,
such that the matrix A and the vectors b and c are nonnegative.
Examples
Covering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms.^{[9]} For example, the LP relaxations of the set packing problem, the independent set problem, and the matching problem are packing LPs. The LP relaxations of the set cover problem, the vertex cover problem, and the dominating set problem are also covering LPs.
Finding a fractional coloring of a graph is another example of a covering LP. In this case, there is one constraint for each vertex of the graph and one variable for each independent set of the graph.
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 = (x_{1}, x_{2}, ... , x_{n}) is primal feasible and that y = (y_{1}, y_{2}, ... , y_{m}) is dual feasible. Let (w_{1}, w_{2}, ..., w_{m}) denote the corresponding primal slack variables, and let (z_{1}, z_{2}, ... , z_{n}) denote the corresponding dual slack variables. Then x and y are optimal for their respective problems if and only if
 x_{j} z_{j} = 0, for j = 1, 2, ... , n, and
 w_{i} y_{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 to zero. Likewise, if the jth slack variable of the dual is not zero, then the jth variable of the primal is equal to zero.
This necessary condition for optimality conveys a fairly simple economic principle. In standard form (when maximizing), if there is slack in a constrained primal resource (i.e., there are "leftovers"), then additional quantities of that resource must have no value. Likewise, if there is slack in the dual (shadow) price nonnegativity constraint requirement, i.e., the price is not zero, then there must be scarce supplies (no "leftovers").
Theory
Existence of optimal solutions
Geometrically, the linear constraints define the feasible region, which is a convex polytope. A 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 ≥ 2 and x ≤ 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 functions (alternatively, by the minimum principle for concave functions) 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 (i.e., the constant function taking the value zero everywhere). For this feasibility problem with the zerofunction for its objectivefunction, 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.
Algorithms
Basis exchange algorithms
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 nondecreasing values of the objective function until an optimum is reached for sure. In many practical problems, "stalling" occurs: many pivots are made with no increase in the objective function.^{[10]}^{[11]} In rare practical problems, the usual versions of the simplex algorithm may actually "cycle".^{[11]} To avoid cycles, researchers developed new pivoting rules.^{[12]}
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,^{[13]} which is similar to its behavior on practical problems.^{[10]}^{[14]}
However, the simplex algorithm has poor worstcase 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.^{[10]}^{[15]}^{[16]} In fact, for some time it was not known whether the linear programming problem was solvable in polynomial time, i.e. of complexity class P.
Crisscross algorithm
Like the simplex algorithm of Dantzig, the crisscross algorithm is a basisexchange algorithm that pivots between bases. However, the crisscross algorithm need not maintain feasibility, but can pivot rather from a feasible basis to an infeasible basis. The crisscross algorithm does not have polynomial timecomplexity for linear programming. Both algorithms visit all 2^{D} corners of a (perturbed) cube in dimension D, the Klee–Minty cube, in the worst case.^{[12]}^{[17]}
Interior point
In contrast to the simplex algorithm, which finds an optimal solution by traversing the edges between vertices on a polyhedral set, interiorpoint methods move through the interior of the feasible region.
Ellipsoid algorithm, following Khachiyan
This is the first worstcase polynomialtime algorithm ever found for linear programming. To solve a problem which has n variables and can be encoded in L input bits, this algorithm runs in [math]\displaystyle{ O(n^6 L) }[/math] time.^{[6]} Leonid Khachiyan solved this longstanding complexity issue in 1979 with the introduction of the ellipsoid method. The convergence analysis has (realnumber) predecessors, notably the iterative methods developed by Naum Z. Shor and the approximation algorithms by Arkadi Nemirovski and D. Yudin.
Projective algorithm of Karmarkar
Khachiyan's algorithm was of landmark importance for establishing the polynomialtime solvability of linear programs. The algorithm was not a computational breakthrough, as the simplex method is more efficient for all but specially constructed families of linear programs.
However, Khachiyan's algorithm inspired new lines of research in linear programming. In 1984, N. Karmarkar proposed a projective method for linear programming. Karmarkar's algorithm^{[7]} improved on Khachiyan's^{[6]} worstcase polynomial bound (giving [math]\displaystyle{ O(n^{3.5}L) }[/math]). Karmarkar claimed that his algorithm was much faster in practical LP than the simplex method, a claim that created great interest in interiorpoint methods.^{[18]} Since Karmarkar's discovery, many interiorpoint methods have been proposed and analyzed.
Vaidya's 87 algorithm
In 1987, Vaidya proposed an algorithm that runs in [math]\displaystyle{ O(n^3) }[/math] time.^{[19]}
Vaidya's 89 algorithm
In 1989, Vaidya developed an algorithm that runs in [math]\displaystyle{ O(n^{2.5}) }[/math] time.^{[20]} Formally speaking, the algorithm takes [math]\displaystyle{ O( (n+d)^{1.5} n L) }[/math] arithmetic operations in the worst case, where [math]\displaystyle{ d }[/math] is the number of constraints, [math]\displaystyle{ n }[/math] is the number of variables, and [math]\displaystyle{ L }[/math] is the number of bits.
Input sparsity time algorithms
In 2015, Lee and Sidford showed that linear programming can be solved in [math]\displaystyle{ \tilde O((nnz(A) + d^2)\sqrt{d}L) }[/math] time,^{[21]} where [math]\displaystyle{ nnz(A) }[/math] represents the number of nonzero elements, and it remains taking [math]\displaystyle{ O(n^{2.5}L) }[/math] in the worst case.
Current matrix multiplication time algorithm
In 2019, Cohen, Lee and Song improved the running time to [math]\displaystyle{ \tilde O( ( n^{\omega} + n^{2.5\alpha/2} + n^{2+1/6} ) L) }[/math] time, [math]\displaystyle{ \omega }[/math] is the exponent of matrix multiplication and [math]\displaystyle{ \alpha }[/math] is the dual exponent of matrix multiplication.^{[22]} [math]\displaystyle{ \alpha }[/math] is (roughly) defined to be the largest number such that one can multiply an [math]\displaystyle{ n \times n }[/math] matrix by a [math]\displaystyle{ n \times n^\alpha }[/math] matrix in [math]\displaystyle{ O(n^2) }[/math] time. In a followup work by Lee, Song and Zhang, they reproduce the same result via a different method.^{[23]} These two algorithms remain [math]\displaystyle{ \tilde O( n^{2+1/6} L ) }[/math] when [math]\displaystyle{ \omega = 2 }[/math] and [math]\displaystyle{ \alpha = 1 }[/math]. The result due to Jiang, Song, Weinstein and Zhang improved [math]\displaystyle{ \tilde O ( n^{2+1/6} L) }[/math] to [math]\displaystyle{ \tilde O ( n^{2+1/18} L) }[/math].^{[24]}
Comparison of interiorpoint methods and simplex algorithms
The current opinion is that the efficiencies of good implementations of simplexbased methods and interior point methods are similar for routine applications of linear programming. However, for specific types of LP problems, it may be that one type of solver is better than another (sometimes much better), and that the structure of the solutions generated by interior point methods versus simplexbased methods are significantly different with the support set of active variables being typically smaller for the latter one.^{[25]}
Open problems and recent work
Unsolved problem in computer science: Does linear programming admit a strongly polynomialtime algorithm? (more unsolved problems in computer science)

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 largescale linear programs.
 Does LP admit a strongly polynomialtime algorithm?
 Does LP admit a strongly polynomialtime algorithm to find a strictly complementary solution?
 Does LP admit a polynomialtime 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 interiorpoint techniques, no algorithms have yet been found that allow strongly polynomialtime 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.
 Are there pivot rules which lead to polynomialtime simplex variants?
 Do all polytopal graphs have polynomially bounded diameter?
These questions relate to the performance analysis and development of simplexlike methods. The immense efficiency of the simplex algorithm in practice despite its exponentialtime 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.
The simplex algorithm and its variants fall in the family of edgefollowing 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 graphtheoretical diameter of polytopal 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 edgefollowing 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, crisscross pivot methods do not preserve (primal or dual) feasibility – they may visit primal feasible, dual feasible or primalanddual infeasible bases in any order. Pivot methods of this type have been studied since the 1970s.^{[26]} 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 polynomialtime crisscross pivot algorithm without resolving questions about the diameter of general polytopes.^{[12]}
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) NPhard. 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 NPhard, and in fact the decision version was one of Karp's 21 NPcomplete 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 NPhard 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 righthand sides of the constraints are integers or – more general – where the system has the total dual integrality (TDI) property.
Advanced algorithms for solving integer linear programs include:
 cuttingplane 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.
Such integerprogramming algorithms are discussed by Padberg and in Beasley.
Integral linear programs
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]\displaystyle{ P = \{x \mid Ax \ge 0\} }[/math] is said to be integral if for all bounded feasible objective functions c, the linear program [math]\displaystyle{ \{\max cx \mid x \in P\} }[/math] has an optimum [math]\displaystyle{ x^* }[/math] with integer coordinates. As observed by Edmonds and Giles in 1977, one can equivalently say that the polyhedron [math]\displaystyle{ P }[/math] is integral if for every bounded feasible integral objective function c, the optimal value of the linear program [math]\displaystyle{ \{\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.
Terminology is not consistent throughout the literature, so one should be careful to distinguish the following two concepts,
 in an integer linear program, described in the previous section, variables are forcibly constrained to be integers, and this problem is NPhard in general,
 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 polynomialsize 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. There are other general methods including the integer decomposition property and total dual integrality. Other specific wellknown integral LPs include the matching polytope, lattice polyhedra, submodular flow polyhedra, and the intersection of two generalized polymatroids/gpolymatroids – e.g. see Schrijver 2003.
Solvers and scripting (programming) languages
Permissive licenses:
Name  License  Brief info 

Gekko  MIT License  Opensource library for solving largescale LP, QP, QCQP, NLP, and MIP optimization 
GLOP  Apache v2  Google's opensource linear programming solver 
JuMP  MPL License  Opensource modeling language with solvers for largescale LP, QP, QCQP, SDP, SOCP, NLP, and MIP optimization 
Pyomo  BSD  An opensource modeling language for largescale linear, mixed integer and nonlinear optimization 
SCIP  Apache v2  A generalpurpose constraint integer programming solver with an emphasis on MIP. Compatible with Zimpl modelling language. 
SuanShu  Apache v2  An opensource suite of optimization algorithms to solve LP, QP, SOCP, SDP, SQP in Java 
Copyleft (reciprocal) licenses:
Name  License  Brief info 

ALGLIB  GPL 2+  An LP solver from ALGLIB project (C++, C#, Python) 
Cassowary constraint solver  LGPL  An incremental constraint solving toolkit that efficiently solves systems of linear equalities and inequalities 
CLP  CPL  An LP solver from COINOR 
glpk  GPL  GNU Linear Programming Kit, an LP/MILP solver with a native C API and numerous (15) thirdparty wrappers for other languages. Specialist support for flow networks. Bundles the AMPLlike GNU MathProg modelling language and translator. 
lp solve  LGPL v2.1  An LP and MIP solver featuring support for the MPS format and its own "lp" format, as well as custom formats through its "eXternal Language Interface" (XLI).^{[27]}^{[28]} Translating between model formats is also possible.^{[29]} 
Qoca  GPL  A library for incrementally solving systems of linear equations with various goal functions 
RProject  GPL  A programming language and software environment for statistical computing and graphics 
MINTO (Mixed Integer Optimizer, an integer programming solver which uses branch and bound algorithm) has publicly available source code^{[30]} but is not open source.
Proprietary licenses:
Name  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 BranchandCut or Column Generation, can also be implemented. The tool calls an appropriate solver such as CPLEX 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 largescale linear, mixed integer and nonlinear optimisation with a free student limited version available (500 variables and 500 constraints). 
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 plugins, including XPRESS, Gurobi, Artelys Knitro, and MOSEK. 
APMonitor  API to MATLAB and Python. Solve example Linear Programming (LP) problems through MATLAB, Python, or a webinterface. 
CPLEX  Popular solver with an API for several programming languages, and also has a modelling language and works with AIMMS, AMPL, GAMS, MPL, OpenOpt, OPL Development Studio, and TOMLAB. Free for academic use. 
Excel Solver Function  A nonlinear solver adjusted to spreadsheets in which function evaluations are based on the recalculating cells. Basic version available as a standard addon for Excel. 
FortMP  
GAMS  
Gurobi Optimizer  
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. 
LINDO  Solver with an API for large scale optimization of linear, integer, quadratic, conic and general nonlinear programs with stochastic programming extensions. It offers a global optimization procedure for finding guaranteed globally optimal solution to general nonlinear programs with continuous and discrete variables. It also has a statistical sampling API to integrate MonteCarlo simulations into an optimization framework. It has an algebraic modeling language (LINGO) and allows modeling within a spreadsheet (What'sBest). 
Maple  A generalpurpose programminglanguage for symbolic and numerical computing. 
MATLAB  A generalpurpose and matrixoriented programminglanguage for numerical computing. Linear programming in MATLAB requires the Optimization Toolbox in addition to the base MATLAB product; available routines include INTLINPROG and LINPROG 
Mathcad  A WYSIWYG math editor. It has functions for solving both linear and nonlinear optimization problems. 
Mathematica  A generalpurpose programminglanguage 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 nonsparse 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 Javabased modeling language for optimization with a free version available.^{[31]}^{[32]} 
SAS/OR  A suite of solvers for Linear, Integer, Nonlinear, DerivativeFree, Network, Combinatorial and Constraint Optimization; the Algebraic modeling language OPTMODEL; and a variety of vertical solutions aimed at specific problems/markets, all of which are fully integrated with the SAS System. 
XPRESS  Solver for largescale linear programs, quadratic programs, general nonlinear and mixedinteger programs. Has API for several programming languages, also has a modelling language Mosel and works with AMPL, GAMS. Free for academic use. 
VisSim  A visual block diagram language for simulation of dynamical systems. 
See also
 Convex programming
 Dynamic programming
 Expected shortfall § Optimization of expected shortfall
 Input–output model
 Job shop scheduling
 Least absolute deviations
 Leastsquares spectral analysis
 Linear algebra
 Linear production game
 Linearfractional programming (LFP)
 LPtype problem
 Mathematical programming
 Nonlinear programming
 Odds algorithm used to solve optimal stopping problems
 Oriented matroid
 Quadratic programming, a superset of linear programming
 Semidefinite programming
 Shadow price
 Simplex algorithm, used to solve LP problems
Notes
 ↑ ^{1.0} ^{1.1} Gerard Sierksma; Yori Zwols (2015). Linear and Integer Optimization: Theory and Practice (3rd ed.). CRC Press. p. 1. ISBN 9781498710169.
 ↑ "Linear programming  Definition & Facts  Britannica" (in en). https://www.britannica.com/science/linearprogrammingmathematics.
 ↑ ^{3.0} ^{3.1} ^{3.2} George B. Dantzig (April 1982). "Reminiscences about the origins of linear programming". Operations Research Letters 1 (2): 43–48. doi:10.1016/01676377(82)900438. https://apps.dtic.mil/sti/pdfs/ADA112060.pdf.
 ↑ Alexander Schrijver (1998). Theory of Linear and Integer Programming. John Wiley & Sons. pp. 221–222. ISBN 9780471982326.
 ↑ ^{5.0} ^{5.1} ^{5.2} Dantzig, George B.; Thapa, Mukund Narain (1997). Linear programming. New York: Springer. p. xxvii. ISBN 0387948333. OCLC 35318475.
 ↑ ^{6.0} ^{6.1} ^{6.2} Leonid Khachiyan (1979). "A Polynomial Algorithm for Linear Programming". Doklady Akademii Nauk SSSR 224 (5): 1093–1096.
 ↑ ^{7.0} ^{7.1} Narendra Karmarkar (1984). "A New PolynomialTime Algorithm for Linear Programming". Combinatorica 4 (4): 373–395. doi:10.1007/BF02579150.
 ↑ M. Grundmann; V. Kwatra; I. Essa (2011). "Autodirected video stabilization with robust L1 optimal camera paths" (in English). CVPR 2011. pp. 225–232. doi:10.1109/CVPR.2011.5995525. ISBN 9781457703942. https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/37041.pdf.
 ↑ (Vazirani 2001)
 ↑ ^{10.0} ^{10.1} ^{10.2} (Dantzig Thapa)
 ↑ ^{11.0} ^{11.1} (Padberg 1999)
 ↑ ^{12.0} ^{12.1} ^{12.2} Fukuda, Komei; Terlaky, Tamás (1997). Thomas M. Liebling. ed. "Crisscross methods: A fresh view on pivot algorithms". Mathematical Programming, Series B 79 (1–3): 369–395. doi:10.1007/BF02614325.
 ↑ (Borgwardt 1987)
 ↑ (Todd 2002)
 ↑ (Murty 1983)
 ↑ (Papadimitriou Steiglitz)
 ↑ Roos, C. (1990). "An exponential example for Terlaky's pivoting rule for the crisscross simplex method". Mathematical Programming. Series A 46 (1): 79–84. doi:10.1007/BF01585729.
 ↑ Strang, Gilbert (1 June 1987). "Karmarkar's algorithm and its place in applied mathematics". The Mathematical Intelligencer 9 (2): 4–10. doi:10.1007/BF03025891. ISSN 03436993.
 ↑ Vaidya, Pravin M. (1987). "An algorithm for linear programming which requires [math]\displaystyle{ {O} (((m+ n) n^2+(m+ n)^{1.5} n) L) }[/math] arithmetic operations". 28th Annual IEEE Symposium on Foundations of Computer Science.
 ↑ Vaidya, Pravin M. (1989). "30th Annual Symposium on Foundations of Computer Science". 30th Annual Symposium on Foundations of Computer Science. pp. 332–337. doi:10.1109/SFCS.1989.63499. ISBN 0818619821.
 ↑ Lee, YinTat; Sidford, Aaron (2015). "Efficient inverse maintenance and faster algorithms for linear programming". FOCS '15 Foundations of Computer Science.
 ↑ Cohen, Michael B.; Lee, YinTat; Song, Zhao (2018). "Solving Linear Programs in the Current Matrix Multiplication Time". 51st Annual ACM Symposium on the Theory of Computing.
 ↑ Lee, YinTat; Song, Zhao; Zhang, Qiuyi (2019). "Solving Empirical Risk Minimization in the Current Matrix Multiplication Time". Conference on Learning Theory.
 ↑ Jiang, Shunhua; Song, Zhao; Weinstein, Omri; Zhang, Hengjie (2020). "Faster Dynamic Matrix Inverse for Faster LPs".
 ↑ Illés, Tibor; Terlaky, Tamás (2002). "Pivot versus interior point methods: Pros and cons". European Journal of Operational Research 140 (2): 170. doi:10.1016/S03772217(02)000619. https://strathprints.strath.ac.uk/9200/.
 ↑ Anstreicher, Kurt M.; Terlaky, Tamás (1994). "A Monotonic BuildUp Simplex Algorithm for Linear Programming". Operations Research 42 (3): 556–561. doi:10.1287/opre.42.3.556. ISSN 0030364X.
 ↑ "lp_solve reference guide (5.5.2.5)". mit.edu. https://web.mit.edu/lpsolve/doc/index.htm.
 ↑ "External Language Interfaces". http://lpsolve.sourceforge.net/5.5/XLI.htm.
 ↑ "lp_solve command". http://lpsolve.sourceforge.net/5.5/lp_solve.htm.
 ↑ "COR@L – Computational Optimization Research At Lehigh". lehigh.edu. http://coral.ie.lehigh.edu/~minto/download.html.
 ↑ http://www.intertrans.eu/resources/Zesch_Hellingrath_2010_Integrated+ProductionDistribution+Planning.pdf OptimJ used in an optimization model for mixedmodel assembly lines, University of Münster
 ↑ http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076 OptimJ used in an Approximate SubgamePerfect Equilibrium Computation Technique for Repeated Games
References
 Kantorovich, L. V. (1940). "Об одном эффективном методе решения некоторых классов экстремальных проблем". Doklady Akad Sci SSSR 28: 211–214.
 F. L. Hitchcock: The distribution of a product from several sources to numerous localities, Journal of Mathematics and Physics, 20, 1941, 224–230.
 G.B Dantzig: Maximization of a linear function of variables subject to linear inequalities, 1947. Published pp. 339–347 in T.C. Koopmans (ed.):Activity Analysis of Production and Allocation, New YorkLondon 1951 (Wiley & ChapmanHall)
 J. E. Beasley, editor. Advances in Linear and Integer Programming. Oxford Science, 1996. (Collection of surveys)
 Bland, Robert G. (1977). "New Finite Pivoting Rules for the Simplex Method". Mathematics of Operations Research 2 (2): 103–107. doi:10.1287/moor.2.2.103.
 Borgwardt, KarlHeinz (1987). The Simplex Algorithm: A Probabilistic Analysis. Algorithms and Combinatorics. 1. SpringerVerlag. (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. SpringerVerlag.
 Dantzig, George B.; Thapa, Mukund N. (2003). Linear Programming 2: Theory and Extensions. SpringerVerlag. (Comprehensive, covering e.g. pivoting and interiorpoint algorithms, largescale problems, decomposition following Dantzig–Wolfe and Benders, and introducing stochastic programming.)
 Edmonds, Jack; Giles, Rick (1977). "A MinMax Relation for Submodular Functions on Graphs". Studies in Integer Programming. Annals of Discrete Mathematics. 1. pp. 185–204. doi:10.1016/S01675060(08)707349. ISBN 9780720407655.
 Fukuda, Komei; Terlaky, Tamás (1997). Thomas M. Liebling. ed. "Crisscross methods: A fresh view on pivot algorithms". Mathematical Programming, Series B 79 (1–3): 369–395. doi:10.1007/BF02614325.
 Gondzio, Jacek; Terlaky, Tamás (1996). "3 A computational view of interior point methods". in J. E. Beasley. Advances in linear and integer programming. Oxford Lecture Series in Mathematics and its Applications. 4. New York: Oxford University Press. pp. 103–144. Postscript file at website of Gondzio and at McMaster University website of Terlaky. http://www.maths.ed.ac.uk/~gondzio/CV/oxford.ps.
 Murty, Katta G. (1983). Linear programming. New York: John Wiley & Sons, Inc.. pp. xix+482. (comprehensive reference to classical approaches). ISBN 9780471097259.
 Evar D. Nering and Albert W. Tucker, 1993, Linear Programs and Related Problems, Academic Press. (elementary)
 Padberg, M. (1999). Linear Optimization and Extensions, Second Edition. SpringerVerlag. (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.)
 Papadimitriou, Christos H.; Steiglitz, Kenneth. Combinatorial Optimization: Algorithms and Complexity (Corrected republication with a new preface ed.). Dover. (computer science)
 Todd, Michael J. (February 2002). "The many facets of linear programming". Mathematical Programming 91 (3): 417–436. doi:10.1007/s101070100261. (Invited survey, from the International Symposium on Mathematical Programming.)
 Vanderbei, Robert J. (2001). Linear Programming: Foundations and Extensions. Springer Verlag.
 Vazirani, Vijay V. (2001). Approximation Algorithms. SpringerVerlag. ISBN 9783540653677. (Computer science)
Further reading
Library resources about Linear programming 
 Dmitris Alevras and Manfred W. Padberg, Linear Optimization and Extensions: Problems and Solutions, Universitext, SpringerVerlag, 2001. (Problems from Padberg with solutions.)
 de Berg, Mark; van Kreveld, Marc; Overmars, Mark; Schwarzkopf, Otfried (2000). Computational Geometry (2nd revised ed.). SpringerVerlag. ISBN 9783540656203. https://archive.org/details/computationalgeo00berg. Chapter 4: Linear Programming: pp. 63–94. Describes a randomized halfplane intersection algorithm for linear programming.
 Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NPCompleteness. W.H. Freeman. ISBN 9780716710455. A6: MP1: INTEGER PROGRAMMING, pg.245. (computer science, complexity theory)
 Gärtner, Bernd; Matoušek, Jiří (2006). Understanding and Using Linear Programming. Berlin: Springer. ISBN 3540306978. (elementary introduction for mathematicians and computer scientists)
 Cornelis Roos, Tamás Terlaky, JeanPhilippe Vial, Interior Point Methods for Linear Optimization, Second Edition, SpringerVerlag, 2006. (Graduate level)
 Alexander Schrijver (2003). Combinatorial optimization: polyhedra and efficiency. Springer.
 Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0471982326 (mathematical)
 Gerard Sierksma; Yori Zwols (2015). Linear and Integer Optimization: Theory and Practice. CRC Press. ISBN 9781498710169.
 Gerard Sierksma; Diptesh Ghosh (2010). Networks in Action; Text and Computer Exercises in Network Optimization. Springer. ISBN 9781441955128. (linear optimization modeling)
 H. P. Williams, Model Building in Mathematical Programming, Fifth Edition, 2013. (Modeling)
 Stephen J. Wright, 1997, PrimalDual InteriorPoint Methods, SIAM. (Graduate level)
 Yinyu Ye, 1997, Interior Point Algorithms: Theory and Analysis, Wiley. (Advanced graduatelevel)
 Ziegler, Günter M., Chapters 1–3 and 6–7 in Lectures on Polytopes, SpringerVerlag, New York, 1994. (Geometry)
External links
 Guidance On Formulating LP Problems
 Mathematical Programming Glossary
 The Linear Programming FAQ
 Benchmarks For Optimisation Software
Original source: https://en.wikipedia.org/wiki/Linear programming.
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