Algorithm of the simplex

The algorithm of the simplex of George Dantzig is an at the same time fundamental and very popular technique for the problems of linear Programming. Thus, being given a whole of linear inequalities on n variable real, the algorithm makes it possible to find the solution optimal for a function objective, which is it also linear (the algorithm still functions when the function is increasing of each one of n variable).

In geometrical terms, the whole of the linear inequalities defines a Polytope in space in n dimensions (polygon in 2 dimensions and polyhedron in 3 dimensions) and it acts to find the top optimal for the performance index given. The idea of the algorithm consists starting from an unspecified top of the polytope and, with each iteration, of going at an adjacent top if it is possible to find the best for the function objective. If there is not, the algorithm stops by concluding that the current top is optimal. In general, there are several adjacent tops at the top running which are better for the objective. It is necessary to select only one of them, the rule of selection is called rule of swivelling .

It was shown for the principal rules of swivelling employed that the algorithm of the simplex could take an exponential computing time. In particular, it is not known if there exists a rule of swivelling which would ensure that the algorithm finishes after a polynomial number of stages.

Nevertheless, the algorithm of the simplex is very effective in practice and it is implemented in all the linear solveurs of programs. There exists inter alia an algorithm based on the Méthode of the ellipsoid, which solves these problems in a polynomial time, but which is in practice often less powerful than the algorithm of the simplex.

See too

External bonds

  • Algorithm Simplex. Detailed illustration of the execution of the algorithm (version “table”).

  • example of the algorithm of simplex. The algorithm of the simplex applied in a very didactic way to an example (version “table”) - file pdf.

Random links:Copa América | Montéhus | Power Macintosh 9600 | Spessa | Gregor Braun | Temelec,_la_Californie