An Approach to Some Structured Linear Programming ProblemsAuthor(s): John M. Bennett and David R. GreenSource: Operations Research, Vol. 17, No. 4 (Jul. - Aug., 1969), pp. 749-750Published by: INFORMSStable URL: http://www.jstor.org/stable/168545 .

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Patrick D. Krolak 749

23. - , Problems Currently Used to Test Integer Linear Programming Algorithms," Technical Report, Information Engineering Dept., Vander- bilt Univ., May 1969.

24. C. E. LEMKE AND K. SPIELBERG, "Direct Search Zero-One and Mixed Integer Programming," Technical Report No. 3, IBM Corporation, New York, September, 1967.

AN APPROACH TO SOME STRUCTURED LINEAR

PROGRAMMING PROBLEMS

John M. Bennett and David R. Green

University of Sydney, Sydney, Australia

(Received October 10, 1966)

A recent paper by J. M. BENNETT describes a decomposition algorithm for the class of linear programming problems commonly called 'angular systems.' This note draws attention to a variant of the algorithm that is particularly suited to problems in which certain submatrices are sparse.

W ̂ . rE CONSIDER the decomposition algorithm for the class of linear program- V Vming problems, commonly called 'angular systems,' that was presented in a

recent paper by J. M. Bennett,(1] and assume that its submatrices Bt are sparse. The result is a variant of the algorithm that is particularly suited to problems where this assumption holds.

This variant involves modifying the calculation of the shadow costs. The method outlined in Bennett's paper [equation (7)] uses a matrix

G= [. *Bi1)-Bi)Ai(2A')) I. 1 (1)

for computing the shadow costs. This matrix is used for no other purpose in the algorithm.

An alternative procedure for computing the shadow costs is to calculate the last row of the transforming matrix T and then to post-multiply this row by

[A11 ~~~~~~~~~~~~~~(2)

Akil)

LB'(') . . . Bk(l)j

The last P elements of the last row of T are given by Fp., and the remaining elements are available as

- [Fp.B1(2)A1(2)I... *Fp.Bk(2)Ak(2)I] (3)

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750 Letters to the Editor

The straightforward procedure of calculating the elements of (3) ab initio at each iteration appears to have as low an operations count as any method for updating (3) from one step to the next.

The reduction in storage requirements achieved by abandoning G is (M -N -

P)P locations, which is a considerable saving for large P. However, this reduction in storage is gained at the expense of an increase in the operations count for the calculations.

Thus, at each step, computing (3) involves

Zk ni2+PN operations,

and the subsequent postmultiplication by (2) yields

A, * (m,-nj-pj)nj+P(M-N-P) operations, giving a total of

E=k (mj-pj)nj+P(M-P) operations.

This is partly offset by the saving during each step of the (M -N -P)P opera- tions previously required to compute Fp.G. In addition, we avoid the operations involved in updating G. These number effectively zero in the case of Type 1 and Type 3 interchanges, but amount to (nj+P)(mj-nj-pj)+njP operations for Type 2 and Type 4 interchanges.

Thus, whichever interchange is involved, the increase in operations count is approximately

'-k (mi-p)ni+PN if k is large.

For the simplified case given in the summary of operations counts in the paper, this total increase becomes approximately kn(m+n) operations. For instance, when k = 50, m = 20, n = P = 10, the number of operations executed in the course of one iteration is about two and a half times greater than when the approach of reference 1 is used, while the reduction in storage experiments if the tableau is being held in full is approximately 17 per cent. If the submatrices are sparse, and use has been made of this sparseness when storing the original tableau, the variant leads to an even higher proportional reduction in storage requirements.

The advantages of this alternative approach for the computation of shadow costs first became apparent during a discussion between one of the authors (J.M.B.) and E. M. BEALE.

REFERENCE

1. J. M. BENNETT, "An Approach to Some Structured Linear Programming Prob- lems," Opns. Res. 14, 636-645 (1966).

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