Simplex algorithm
![A system of linear inequalities defines a polytope as a feasible region. The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex
of the optimum solution.](/Images/godic/202502/11/Simplex-description-en.svg0922.png")
![Polyhedron of simplex algorithm in 3D](/Images/godic/202502/11/Simplex-method-3-dimensions0922.png")
![The graph of a strictly concave quadratic function is shown in blue, with its unique maximum shown as a red dot. Below the graph appears the contours of the function: The level sets are nested ellipses.](/Images/godic/202502/11/Max_paraboloid.svg0922.png")
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the twentieth century.
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial cones, and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function.