Stepwise procedures using both directions
Pages 121.401 - 121.404
Abstract
Suppose S is the set of integers (1,2,...,m) and for each h there is a real-valued function fh (il,...,ih) of h arguments from S. The combinatorial problem posed is that of finding for each h that subset Sh of S with h elements which optimizes (maximizes or minimizes) fh from among all subsets of h integers of S.
Example 1. Multiple regression. Let y = co + clxl + ... + cmxm be the equation used in a linear regression based on p observations of the variable y on the variables xl,...,xm. Let fh (il,...,ih) = fh (Sh) be the residual sum of squares of the least squares fit for which cj = O if j is not in Sh. We wish to minimize fh for each h. Sometimes co is prescribed to be O.
Example 2. Let Ai be m real n-dimensional vectors, i.e., Ai=(ail,...,ain) (i = l,...,m); and let fh (il.,.ih) be the volume of the h-dimensibnal parallelepiped with sides Ail .,.Aih. For each h we wish to maximize fh.
One method of solution is exhaustive - to evaluate the fh for all subsets of S; in all there are 2m1, and for m = 20 this is over a million. However, for Example 1 there is a SHARE program for m @@@@ 10. For large m, an exhaustive procedure is impractical
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Association for Computing Machinery
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Publication History
Published: 01 January 1961
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