For the retronym referring to television broadcasting, see Broadcast programming.
A pictorial representation of a simple linear program with two variables and six inequalities. The set of feasible solutions is depicted in yellow and forms a polygon, a 2-dimensional polytope. The optimum of the linear cost function is where the red line intersects the polygon. The red line is a level set of the cost function, and the arrow indicates the direction in which we are optimizing.A closed feasible region of a problem with three variables is a convex polyhedron. The surfaces giving a fixed value of the objective function are planes (not shown). The linear programming problem is to find a point on the polyhedron that is on the plane with the highest possible value.
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear 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 polytope. 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
Here the components of are the variables to be determined, and are given vectors, and is a given matrix. The function whose value is to be maximized ( in this case) is called the objective function. The constraints and 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.
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and objective are represented by linear relationships. Linearprogramming is a special case of mathematical programming (also known as mathematical optimization)...
mixed-integer programming problem. In integer linearprogramming, the canonical form is distinct from the standard form. An integer linearprogram in canonical...
The dual of a given linearprogram (LP) is another LP that is derived from the original (the primal) LP in the following schematic way: Each variable in...
mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints are not linear equalities or the objective...
Successive LinearProgramming (SLP), also known as Sequential LinearProgramming, is an optimization technique for approximately solving nonlinear optimization...
The GNU LinearProgramming Kit (GLPK) is a software package intended for solving large-scale linearprogramming (LP), mixed integer programming (MIP),...
function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming. "Programming" in this context refers...
"Linear genetic programming" is unrelated to "linearprogramming". Linear genetic programming (LGP) is a particular method of genetic programming wherein...
ISBN 978-0-538-49790-9 Leonid N. Vaserstein (2006), "LinearProgramming", in Leslie Hogben, ed., Handbook of Linear Algebra, Discrete Mathematics and Its Applications...
4 Linearprogramming problems are the simplest convex programs. In LP, the objective and constraint functions are all linear. Quadratic programming are...
non-negative. In pure integer programming problems, the feasible set is the set of integers (or some subset thereof). In linearprogramming problems, the feasible...
Semidefinite programming (SDP) is a subfield of mathematical programming concerned with the optimization of a linear objective function (a user-specified...
in a diverse range of SQP methods. Sequential linearprogramming Sequential linear-quadratic programming Augmented Lagrangian method SQP methods have been...
Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical...
(such as linear television and linear channels). With the beginning of scheduled television in 1936, television programming was initially only concerned...
optimization, the fundamental theorem of linearprogramming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal...
formulation is widely used in stochastic programming. The general formulation of a two-stage stochastic programming problem is given by: min x ∈ X { g ( x...
Linear actuator Linear element Linear foot Linear system LinearprogrammingLinear differential equation Bilinear Multilinear Linear motor Linear interpolation...
algorithm, an algorithm for solving linearprogramming problems, and for his other work with linearprogramming. In statistics, Dantzig solved two open...
In the theory of linearprogramming, a basic feasible solution (BFS) is a solution with a minimal set of non-zero variables. Geometrically, each BFS corresponds...
mathematical programming problem (a term not directly related to computer programming, but still in use for example in linearprogramming – see History...
theory and coding theory, linearprogramming decoding (LP decoding) is a decoding method which uses concepts from linearprogramming (LP) theory to solve decoding...
simplex algorithm (or simplex method) is a popular algorithm for linearprogramming. The name of the algorithm is derived from the concept of a simplex...
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