In mathematical optimization, Bland's rule (also known as Bland's algorithm, Bland's anti-cycling rule or Bland's pivot rule) is an algorithmic refinement of the simplex method for linear optimization.
With Bland's rule, the simplex algorithm solves feasible linear optimization problems without cycling.[1][2][3]
The original simplex algorithm starts with an arbitrary basic feasible solution, and then changes the basis in order to decrease the minimization target and find an optimal solution. Usually, the target indeed decreases in every step, and thus after a bounded number of steps an optimal solution is found. However, there are examples of degenerate linear programs, on which the original simplex algorithm cycles forever. It gets stuck at a basic feasible solution (a corner of the feasible polytope) and changes bases in a cyclic way without decreasing the minimization target.
Such cycles are avoided by Bland's rule for choosing a column to enter and a column to leave the basis.
Bland's rule was developed by Robert G. Bland, now an Emeritus Professor of operations research at Cornell University, while he was a research fellow at the Center for Operations Research and Econometrics in Belgium.[1]