# A Brief History of Linear and Mixed-Integer Programming Computation

@inproceedings{Bixby2012ABH, title={A Brief History of Linear and Mixed-Integer Programming Computation}, author={Robert E. Bixby}, year={2012} }

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the…

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## References

SHOWING 1-10 OF 45 REFERENCES

The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results

- Computer ScienceMath. Program.
- 1977

An attempt to provide a powerful mathematical programming language, allowing an easy programming of specific studies on medium-size models such as the recursive use of LP or the build-up of algorithms based on the simplex method is described.

Solving Large-Scale Zero-One Linear Programming Problems

- Computer ScienceOper. Res.
- 1983

The results indicate that cutting-planes related to the facets of the underlying polytope are an indispensable tool for the exact solution of this class of problem.

Practical Solution of Large Mixed Integer Programming Problems with Umpire

- Computer Science
- 1974

In this paper we discuss some branch and bound methods implemented in the UMPIRE mathematical programming system for solving practical integer programming problems and give details of computational…

Solving Real-World Linear Programs: A Decade and More of Progress

- EconomicsOper. Res.
- 2002

One person's perspective on the development of computational tools for linear programming is described, followed by historical remarks covering the some 40 years of linear-programming developments that predate my own involvement in this subject.

Markowitz and Manne + Eastman + Land and Doig = Branch and Bound

- Computer Science
- 2012

The branch-and-bound method consists of the repeated application of a process for splitting a space of solutions into two or more subspaces and adopting a bounding mechanism to indicate if it is…

An Automatic Method for Solving Discrete Programming Problems

- History50 Years of Integer Programming
- 2010

In the late 1950s there was a group of teachers and research assistants at the London School of Economics interested in linear programming and its extensions, in particular Helen Makower, George…

Experiments in mixed-integer linear programming

- Computer ScienceMath. Program.
- 1971

The heuristic rules for generating the tree, which are the main features of the method, are presented and numerous parameters allow the user for adjusting the search strategy to a given problem.

SCIP: solving constraint integer programs

- Computer ScienceMath. Program. Comput.
- 2009

An overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs is given and experimental results show that the approach outperforms current state-of-the-art techniques for proving the validity of properties on circuits containing arithmetic.

A new polynomial-time algorithm for linear programming

- Mathematics, Computer ScienceComb.
- 1984

It is proved that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property: the ratio of the radius of the smallest sphere with center a′, containingP′ to theradius of the largest sphere withCenter a′ contained inP′ isO(n).

Solving Mixed Integer Programming Problems Using Automatic Reformulation

- BusinessOper. Res.
- 1987

The solution to optimality of 18 medium-to large-size problems, including production planning problems with setup costs and capacity constraints, multilevel distribution planning problems, drainage and heating system design problems, and electricity generator scheduling problems is reported on.