linear programming

A linear programming problem, or LP, is a problem of optimizing a given linear objective function over some polyhedron. The standard maximization LP, sometimes called the primal problem, is

maximize cTx
s.t. Axb (P)

Here cTx is the objective function and the remaining conditions define the polyhedron which is the feasible region over which the objective function is to be optimized. The dual of ((P)) is the LP

minimize yTb
s.t. yTAcT (D)

The linear constraints for a linear programming problems define a convex polyhedron, called the feasible region for the problem. The weak duality theorem states that if x^ is feasible (i.e. lies in the feasible region) for ((P)) and y^ is feasible for ((D)), then cTx^y^Tb. This follows readily from the above:


The strong duality theorem states that if both LPs are feasible, then the two objective functions have the same optimal value. As a consequence, if either LP has unbounded objective function value, the other must be infeasible. It is also possible for both LP to be infeasible.

The simplex methodMathworldPlanetmath ( of G. B. Dantzig is the algorithmMathworldPlanetmath most commonly used to solve LPs; in practice it runs in polynomial timeMathworldPlanetmath, but the worst-case running time is exponential. Two polynomial-time algorithms for solving LPs are the ellipsoid method of L. G. Khachian and the interior-point method of N. Karmarkar.


  • 1 Chvátal, V., Linear programming, W. H. Freeman and Company, 1983.
  • 2 Cormen, T. H., Leiserson, C. E., Rivest, R. L., and C. Stein, Introduction to algorithms, MIT Press, 2001.
  • 3 Korte, B. and J. Vygen, Combinatorial optimization: theory and algorithms, Springer-Verlag, 2002.
Title linear programming
Canonical name LinearProgramming
Date of creation 2013-03-22 13:41:41
Last modified on 2013-03-22 13:41:41
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Topic
Classification msc 90C05
Synonym LP
Related topic DualityInMathematics
Defines linear programming problem
Defines objective function
Defines feasible region
Defines feasible