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

@article{Fukuda1997CrisscrossMA, title={Criss-cross methods: A fresh view on pivot algorithms}, author={Komei Fukuda and Tam{\'a}s Terlaky}, journal={Mathematical Programming}, year={1997}, volume={79}, pages={369-395} }

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…

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## References

SHOWING 1-10 OF 96 REFERENCES

### Some generalizations of the criss-cross method for quadratic programming

- Mathematics
- 1992

Three generalizations of the criss-cross method for quadratic programming are presented here. Tucker’s, Cottle’s and Dantzig’s principal pivoting methods are specialized as diagonal and exchange…

### New variants of finite criss-cross pivot algorithms for linear programming

- Computer ScienceEur. J. Oper. Res.
- 1999

### The Criss-Cross Method for Solving Linear Programming Problems

- Computer Science
- 1969

The Criss-Cross Method of solving linear programming problems, a primal-dual scheme, normally begins with a problem solution that is neither primal nor dual feasible, and generates an optimal feasible solution in a finite number of iterations.

### A new proof for the criss-cross method for quadratic programming

- Mathematics
- 1992

In a working paper Chang has introduced a simple method for the linear complementarity problem involving a nonnegative definite matrix, based on principal pivoting and the least-index rule of Bland.…

### A convergent criss-cross method

- Mathematics
- 1985

Our paper presents a new Criss-Cross method for solving linear programming problems. Starting from a neither primal nor dual feasible solution, we reach an optimal solution in finite number of steps…

### Pivot rules for linear programming: A survey on recent theoretical developments

- Mathematics, Computer ScienceAnn. Oper. Res.
- 1993

The various pivot rules of the simplex method and its variants that have been developed in the last two decades are discussed, starting from the appearance of the minimal index rule of Bland.

### An exponential example for Terlaky's pivoting rule for the criss-cross simplex method

- MathematicsMath. Program.
- 1990

It is shown that the required number of iterations may be exponential in the number of variables and constraints of the problem.

### The existence of a short sequence of admissible pivots to an optimal basis in LP and LCP

- Mathematics
- 1997

We say an LP (linear programming) is fully nondegenerate if both the primal and the dual problems are nondegenerate. In this paper, we prove the existence of a sequence of |B∖B| admissible pivot from…