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}
}
  • K. FukudaT. Terlaky
  • Published 1 October 1997
  • Economics, Computer Science
  • Mathematical Programming
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… 

On the existence of a short pivoting sequence for a linear program

New variants of the criss-cross method for linearly constrained convex quadratic programming

A non-monotonic infeasible interior-exterior point algorithm for linear programming.

A variant of the exterior point algorithmic family for the linear problem, iEPSA, is presented in an attempt to shed light upon this ambiguity and extrapolate a significant implication that in some cases, a combination of interior-exterior methods is considerably more efficient.

Pivot versus interior point methods: Pros and cons

Anstreicher–Terlaky type monotonic simplex algorithms for linear feasibility problems

A new monotonic build-up (MBU) simplex algorithm for linear feasibility problems and a new recursive procedure to handle strongly degenerate problems as well are constructed.

Closing the gap in pivot methods for linear programming

Pivot methods that solve linear programs by trying to close the duality gap from both ends by exploiting the adjacency neighborhood of the primal-dual vertex pair corresponding to B in the underlying primal and dual hyperplane arrangements are proposed.

Interior-point algorithm for sufficient LCPs based on the technique of algebraically equivalent transformation

We present a short-step interior-point algorithm (IPA) for sufficient linear complementarity problems (LCPs) based on a new search direction. An algebraic equivalent transformation (AET) is used on

Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

Dantzig's simplex method searches the graph of the polyhedron, from a vertex of the one-skeleton to a better neighboring one according to some pivot rule, which selects an improving neighbor.
...

References

SHOWING 1-10 OF 96 REFERENCES

Some generalizations of the criss-cross method for quadratic programming

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

The Criss-Cross Method for Solving Linear Programming Problems

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

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

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

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

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

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
...