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 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 real -valued affine linear function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the smallest or largest value if such a point exists. Linear programs are problems that can be expressed in canonical form as. The expression to be maximized or minimized is called the objective function c T x in this case. In this context, two vectors are 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. 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. The problem of solving a system of linear inequalities dates back at least as far as Fourier , who in published a method for solving them,  and after whom the method of Fourier—Motzkin elimination is named. In a linear programming formulation of a problem that is equivalent to the general linear programming problem was given by the Soviet economist Leonid Kantorovich , who also proposed a method for solving it.
Koopmans formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the Nobel prize in economics. During —, George B. Dantzig independently developed general linear programming formulation to use for planning problems in the US Air Force . In , Dantzig also invented the simplex method that for the first time efficiently tackled the linear programming problem in most cases .
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 .
Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 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 ,  but a larger theoretical and practical breakthrough in the field came in when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems.
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. 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 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 issues.
Although the modern management issues are ever-changing, most companies would like to maximize profits and minimize costs with limited resources. Therefore, many issues can be characterized as linear programming problems. Standard form is the usual and most intuitive form of describing a linear programming problem.
It consists of the following three parts:. The problem is usually expressed in matrix form , and then becomes:. 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.
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:.
Linear programming problems can be converted into an augmented form in order to apply the common form of the simplex algorithm. This form introduces non-negative slack variables to replace inequalities with equalities in the constraints.
The problems can then be written in the following block matrix form:.
Duality in linear programming
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:.
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.
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. A covering LP is a linear program of the form:.
The dual of a covering LP is a packing LP , a linear program of the form:. Covering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms.
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.
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:. Then x and y are optimal for their respective problems if and only if.
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So if the i -th slack variable of the primal is not zero, then the i -th variable of the dual is equal to zero. Likewise, if the j -th slack variable of the dual is not zero, then the j -th 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.
Likewise, if there is slack in the dual shadow price non-negativity constraint requirement, i. Geometrically, the linear constraints define the feasible region , which is a convex polyhedron. 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. 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.
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 that is, 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.
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. The simplex algorithm , developed by George Dantzig in , 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, " stalling " occurs: many pivots are made with no increase in the objective function. 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. 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. 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 polynomial time-complexity for linear programming.
In contrast to the simplex algorithm, which finds an optimal solution by traversing the edges between vertices on a polyhedral set, interior-point methods move through the interior of the feasible region.
This is the first worst-case polynomial-time algorithm ever found for linear programming.
To solve a problem which has n variables and can be encoded in L input bits, this algorithm uses O n 4 L pseudo-arithmetic operations on numbers with O L digits.
Leonid Khachiyan solved this long-standing complexity issue in with the introduction of the ellipsoid method. The convergence analysis has real-number predecessors, notably the iterative methods developed by Naum Z. Shor and the approximation algorithms by Arkadi Nemirovski and D.