Here 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 is the LP
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 is feasible (i.e. lies in the feasible region) for and is feasible for , then . 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 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.
|Date of creation||2013-03-22 13:41:41|
|Last modified on||2013-03-22 13:41:41|
|Last modified by||mathcam (2727)|
|Defines||linear programming problem|