Linear programming, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function over a convex polyhedron specified by linear and non-negativity constraints. Simplistically, linear programming is the optimization of an outcome based on some set of constraints using a linear mathematical model.

Linear programming is implemented in the Wolfram Language as `LinearProgramming`[*c*,

*m*, *b*], which finds a vector which minimizes

the quantity subject to the constraints and for .

Linear programming theory falls within convex optimization theory and is also considered to be an important part of operations

research. Linear programming is extensively used in business and economics, but

may also be used to solve certain engineering problems.

Examples from economics include Leontief’s input-output model, the determination of shadow prices, etc., an example of a business application would be maximizing profit in a factory that manufactures a number of different products from the same raw material using the same resources, and example engineering applications include Chebyshev approximation and the design of structures (e.g., limit analysis of a planar truss).

Linear programming can be solved using the simplex method (Wood and Dantzig 1949, Dantzig 1949) which runs along polytope

edges of the visualization solid to find the best answer. Khachian (1979) found

a polynomial

time algorithm. A much more efficient polynomial

time algorithm was found by Karmarkar (1984). This

method goes through the middle of the solid (making it a so-called interior

point method), and then transforms and warps. Arguably, interior point methods

were known as early as the 1960s in the form of the barrier function methods, but

the media hype accompanying Karmarkar’s announcement led to these methods receiving

a great deal of attention.

Linear programming in which variables may take on integer values only is known as

integer programming.

In the Season 4 opening episode “Trust Metric” (2007) of the television crime drama NUMB3RS, math genius Charlie

Eppes uses the phrase “you don’t need Karmarkar’s algorithm” to mean “you

don’t need to be a rocket scientist to know….”

SEE ALSO: Criss-Cross Method, Ellipsoidal Calculus,

Integer

Programming, Interior Point Method, Kuhn-Tucker Theorem,

Lagrange

Multiplier, Nonlinear Programming, Operations Research, Optimization,

Optimization Theory,

Stochastic

Optimization, Vertex Enumeration

*Portions of this entry contributed by James*

Noyes

REFERENCES:

Bellman, R. and Kalaba, R. *Dynamic*

Programming and Modern Control Theory. New York: Academic Press, 1965.

Dantzig, G. B. “Programming of Interdependent Activities. II. Mathematical

Model.” *Econometrica* **17**, 200-211, 1949.

Dantzig, G. B. *Linear*

Programming and Extensions. Princeton, NJ: Princeton University Press, 1963.

Garey, M. R. and Johnson, D. S. *Computers and Intractability: A Guide to the Theory of NP-Completeness.* New York: W. H.

Freeman, pp. 155-158, 287-288, and 339, 1983.

Greenberg, H. J. “Mathematical Programming Glossary.” https://carbon.cudenver.edu/~hgreenbe/glossary/.

Karloff, H. *Linear*

Programming. Boston, MA: Birkhäuser, 1991.

Khachian, L. G. “A Polynomial Algorithm in Linear Programming.” *Dokl. Akad. Nauk SSSR* **244**, 1093-1096, 1979. English translation in *Soviet*

Math. Dokl. **20**, 191-194, 1979.

Karmarkar, N. “A New Polynomial-Time Algorithm for Linear Programming.”

*Combinatorica* **4**, 373-395, 1984.

Pappas, T. “Projective Geometry & Linear Programming.” *The Joy of Mathematics.* San Carlos, CA: Wide World Publ./Tetra, pp. 216-217,

1989.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Linear Programming and the Simplex Method.” §10.8 in *Numerical*

Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:

Cambridge University Press, pp. 423-436, 1992.

Sultan, A. *Linear Programming: An Introduction with Applications.* San Diego, CA: Academic Press,

1993.

Tokhomirov, V. M. “The Evolution of Methods of Convex Optimization.”

*Amer. Math. Monthly* **103**, 65-71, 1996.

Weisstein, E. W. “Books about Linear Programming.” https://www.ericweisstein.com/encyclopedias/books/LinearProgramming.html.

Wood, M. K. and Dantzig, G. B. “Programming of Interdependent Activities.

I. General Discussion.” *Econometrica* **17**, 193-199, 1949.

Yudin, D. B. and Nemirovsky, A. S. *Problem*

Complexity and Method Efficiency in Optimization. New York: Wiley, 1983.

Referenced on Wolfram|Alpha: Linear Programming

CITE THIS AS:

Noyes, James and Weisstein, Eric W. “Linear Programming.” From *MathWorld*–A

Wolfram Web Resource. https://mathworld.wolfram.com/LinearProgramming.html