linear programming

A linear programming problem, or LP,is a problem of optimizing a givenlinear objective function over some polyhedron. The standardmaximization LP, sometimes called theprimal problem, is

maximize$c^{T}x$

s.t.$Ax\le b$(P)

$x\ge 0$

Here $c^{T}x$ is the objective function and the remaining conditionsdefine the polyhedron which is the feasible region over which theobjective function is to be optimized. The dual of$(\mathit{\text{<span class="ltx\_text ltx\_ref\_tag">(P)</span>}})$ is the LP

minimize$y^{T}b$

s.t.$y^{T}A\ge c^T$(D)

$y\ge 0$

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 $\widehat{x}$ is feasible (i.e. lies in the feasible region) for$(\mathit{\text{<span class="ltx\_text ltx\_ref\_tag">(P)</span>}})$ and $\widehat{y}$ is feasible for $(\mathit{\text{<span class="ltx\_text ltx\_ref\_tag">(D)</span>}})$,then $c^T\widehat{x}\le {\widehat{y}}^{T}b$. This follows readily from the above:

$$c^T\widehat{x}\le ({\widehat{y}}^{T}A)\widehat{x}={\widehat{y}}^T(A\widehat{x})\le y^{T}b.$$

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

The simplex method (http://planetmath.org/SimplexAlgorithm) of G. B. Dantzig isthe algorithmmost commonly used to solve LPs; in practice it runs in polynomial time,but the worst-case running time is exponential. Two polynomial-timealgorithms for solving LPs are the ellipsoid method of L. G. Khachianand the interior-point method of N. Karmarkar.