Criss-cross methods: A fresh view on pivot algorithms

This paper surveys the origins and implications of ( nite) criss-cross methods in linear programming and related problems. Most pivot algorithms, like Dantzig's celebrated Simplex method, need a feasible basis to start with; when the basis is changed, feasibility is preserved until optimality or unboundedness is detected. To obtain a feasible basis a so-called rst phase problem must be solved. On the contrary, a criss-cross method is a pivot algorithm that solves the linear programming problem in one phase without preserving possible feasibility of the intermediate solutions. Getting free from the necessity of preserving feasibility and monotonicity needs a fresh view on algorithmic concepts, it raises new questions and opens new perspectives. After a brief introduction and historical review a nite criss-cross algorithm is presented. Via alternative niteness proofs the exibility of the method is explored. The niteness of this simple one-phase procedures provides us with a most simple algorithmic proof of the strong duality theorem of LP as well. A recent result on the existence of a short pivot path to an optimal basis is given, indicating a certain research direction for designing polynomial pivot algorithms if they ever exist. Some extensions to convex quadratic programming, linear complementarity, fractional linear programming problems and to oriented matroid programming are discussed. Notes on existentially polytime (EP) theorems for linear complementarity problems are included. Some conclusions and notes on possible future research directions close the paper.