Mathematical Programming

, Volume 79, Issue 1–3, pp 369–395

# Criss-cross methods: A fresh view on pivot algorithms

• Komei Fukuda
• Tamás Terlaky
Article

## Abstract

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon.

## Keywords

Linear programming Quadratic programming Linear complementarity problems Oriented matroids Pivot rules Criss-cross method Cycling Recursion

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© The Mathematical Programming Society, Inc 1997

## Authors and Affiliations

• Komei Fukuda
• 1
• 2
• Tamás Terlaky
• 3
1. 1.Department of MathematicsSwiss Federal Institute of TechnologyLausanneSwitzerland
2. 2.Institute for Operations Research, Swiss Federal Institute of TechnologyZürichSwitzerland
3. 3.Faculty of Technical Mathematics and InformaticsDelft University of TechnologyDelftThe Netherlands