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	$\displaystyle c^{T}x$
s.t.	$\displaystyle Ax\leq b{}$	(P)
	$\displaystyle x\geq 0$

Here $c^{T}x$ 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 $({\ref{eq:primal}})$ is the LP

minimize	$\displaystyle y^{T}b$
s.t.	$\displaystyle y^{T}A\geq c^{T}{}$	(D)
	$\displaystyle y\geq 0$

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 $\hat{x}$ is feasible (i.e. lies in the feasible region) for $(\ref{eq:primal})$ and $\hat{y}$ is feasible for $(\ref{eq:dual})$ , then $c^{T}\hat{x}\leq\hat{y}^{T}b$ . This follows readily from the above:

$c^{T}\hat{x}\leq(\hat{y}^{T}A)\hat{x}=\hat{y}^{T}(A\hat{x})\leq y^{T}b.$

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 method (http://planetmath.org/SimplexAlgorithm) of G. B. Dantzig is the algorithm most commonly used to solve LPs; in practice it runs in polynomial time, 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.

References

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