# Simplex method (algorithm)

**DOI:**https://doi.org/10.1007/1-4020-0611-X_953

A computational procedure for solving linear-programming problems of the form: Minimize (maximize) * cx,*v subject to

*v =*

**Ax***v >*

**b, x****0,**where

*v is an*

**A***m*×

*n*matrix,

*v is an*

**c***n*-dimensional row vector,

*v is an*

**b***m*-dimensional column vector, and

*v is an*

**x***n*-dimensional variable vector. The simplex method was developed by George B. Dantzig in the late 1940s. The method starts with a known basic feasible solution or an artificial basic solution, and, given that the problem is feasible, finds a sequence of basic feasible solutions (extreme-point solutions) such that the value of the objective function improves or does not degrade. Under a nondegeneracy assumption, the simplex algorithm will converge in a finite number of steps, as there are only a finite number of extreme points and extreme directions of the underlying convex set of solutions. At most,

*m*variables can be in the solution at a positive level. In each step (iteration) of the simplex method, a new basis is found and developed by...