Bland's rule

In mathematical optimization, Bland's rule (also known as Bland's algorithm or Bland's anti-cycling 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. There are examples of degenerate linear optimization problems on which the original simplex algorithm would cycle forever. Such cycles are avoided by Bland's rule for choosing a column to enter the basis.

Bland's rule was developed by Robert Gary Bland, now a professor of operations research at Cornell University.

Algorithm

One uses Bland's rule during an iteration of the simplex method to decide first what column (known as the entering column) and then row (known as the leaving row) in the tableau to pivot on. Assuming that the problem is to minimize the objective function, the algorithm is loosely defined as follows:

1. Choose the lowest-numbered (i.e., leftmost) nonbasic column t with a positive cost.
2. Now among the rows choose the one with the lowest ratio between the cost and the index in the matrix where the index is greater than zero. If several rows have the same minimum ratio, choose the first one (i.e., topmost) of them.

Further reading

• Bland, Robert G. (May 1977). "New finite pivoting rules for the simplex method". Mathematics of Operations Research 2 (2): 103–107. doi:10.1287/moor.2.2.103. JSTOR 3689647. MR459599. 
• George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. Springer-Verlag.
• Fukuda, Komei; Terlaky, Tamás (1997). Thomas M. Liebling and Dominique de Werra. ed. "Criss-cross methods: A fresh view on pivot algorithms". Mathematical Programming: Series B (Amsterdam: North-Holland Publishing Co.) 79 (Papers from the 16th International Symposium on Mathematical Programming held in Lausanne, 1997): 369–395. doi:10.1016/S0025-5610(97)00062-2. MR1464775. http://www.cas.mcmaster.ca/~terlaky/files/crisscross.ps. 
• Kattta G. Murty, Linear Programming, Wiley, 1983.
• Evar D. Nering and Albert W. Tucker, 1993, Linear Programs and Related Problems, Academic Press.
• M. Padberg, Linear Optimization and Extensions, Second Edition, Springer-Verlag, 1999.
• Christos H. Papadimitriou and Kenneth Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Corrected republication with a new preface, Dover. (computer science)
• Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0-471-98232-6 (mathematical)
• Michael J. Todd (February 2002). "The many facets of linear programming". Mathematical Programming 91 (3).  (Invited survey, from the International Symposium on Mathematical Programming.)