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Combinatorial Optimization - Cristofides N.

Cristofides N. Combinatorial Optimization - Wiley publishing , 2012. - 212 p.
Download (direct link): —Āombinatorialoptimi2012.pdf
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8 Ophelie SIA CDC 6600 1972
9 Sciconic Scicon Univac 1100 Series 1976
10 Tempo Burroughs B7000 and B8000 Series 1975
11 XDLA ICL 1900 Series 1970
In the early 1970ís, codes became available which could handle larger problems. Published computational experience distinguished between two generic types of problems. Type one problems are fundamentally large lpís with a few integer variables, and type two problems are highly structured ilpís. Type one problems are more common in practice, and different strategies are generally used for the two types. Much of the published computational experience relates the specific strategies used for handling the different type problems and is too detailed to relate here. We will relate only a few data points for some of these codes, and refer the reader to the individual articles for greater detail or to Geoffrion and Marsten (1972) or Garfinkel and Nemhauser (1972a) for additional detail. ^
An early version of Code 8 was described by Roy, Benayoun, and Tergny
(1970). Their rules were based to a large extent on penalties. They solve, for example, a problem with 1244 constraints and 3908 variables, 24 of which are integer, in about six minutes of CDC-6600 time. Code 5, based on similar concepts, is described by Benichou et al. (1971). A problem with 721 constraints and 1156 variables, 39 of which are integer, is solved in 18 minutes on the IBM 360/75. Tomlin (1970) also describes a similar algorithm and indicates that incorporation of the penalty (1.10) yields reduction in running times of up to 50 percent over an algorithm based on the penalties (1.7)ó(1.9).
In the middle 1970ís it was discovered that penalty-based codes did not perform well for large problems. This discovery led to experimentation with pseudo costs. It is stressed in Forrest, Hirst, and Tomlin (1974) and Gauthier and Ribiere (1977) that the branching strategy is the key element of a branch and bound code and that pseudo costs are good tools for making these decisions. Both of these articles give detailed experience contrasting different strategies.
Branch and Bound Methods for Integer Programming
19
A good deal of experience is also reported in Breu and Burdet (1974) for
binary milpís.
References
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Beale, E. M. L. and J. A. Tomlin (1970). Special Facilities in a General Mathematical Programming System for Nonconvex Problems Using Ordered Sets of Variables. In Proc. Fifth Int. Conf. on Operational Research, pp. 447-454.
Benichou, M. et al. (1971). Experiments in Mixed-Integer Linear Programming. Mathematical Programming, 1, 76-94.
Breu, R. and C. A. Burdet (1974). Branch and Bound Experiments in Zero-One Programming. Mathematical Programming Study, 2, 1-50.
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Ciarfinkel, R. S. and G. L. Nemhauser (1972a). Integer Programming, Wiley, New ^ York.
Garfinkel, R. S. and G. L. Nemhauser (1972b). Optimal Set Covering: A Survey. In A. M. Geoffrion (Ed.), Perspectives in Optimization, Addison-Wesley. pp. 164-183.
Gauthier, J. M. and F. Ribiere (1977). Experiments in Mixed-Integer Linear Programming Using Pseudo-Costs. Math. Prog, 12, 26-47.
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