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Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming

Published online by Cambridge University Press:  01 January 2025

Michael J. Brusco  and
Stephanie Stahl
Michael J. Brusco*
Affiliation:
Florida State University, Tallahassee, FL, USA
Stephanie Stahl
Affiliation:
Tallahassee, FL, USA
*
Requests for reprints should be sent to: Michael J. Brusco, Department of Marketing, College of Business, Florida State University, Tallahassee, FL 32306-1110, USA. E-mail: mbrusco@cob.fsu.edu
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Abstract

There are two well-known methods for obtaining a guaranteed globally optimal solution to the problem of least-squares unidimensional scaling of a symmetric dissimilarity matrix: (a) dynamic programming, and (b) branch-and-bound. Dynamic programming is generally more efficient than branch-and-bound, but the former is limited to matrices with approximately 26 or fewer objects because of computer memory limitations. We present some new branch-and-bound procedures that improve computational efficiency, and enable guaranteed globally optimal solutions to be obtained for matrices with up to 35 objects. Experimental tests were conducted to compare the relative performances of the new procedures, a previously published branch-and-bound algorithm, and a dynamic programming solution strategy. These experiments, which included both synthetic and empirical dissimilarity matrices, yielded the following findings: (a) the new branch-and-bound procedures were often drastically more efficient than the previously published branch-and-bound algorithm, (b) when computationally feasible, the dynamic programming approach was more efficient than each of the branch-and-bound procedures, and (c) the new branch-and-bound procedures require minimal computer memory and can provide optimal solutions for matrices that are too large for dynamic programming implementation.

Keywords

combinatorial data analysisleast-squares unidimensional scalingbranch-and-bounddynamic programming
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 2 , June 2005 , pp. 253 - 270
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

The authors gratefully acknowledge the helpful comments of three anonymous reviewers and the Editor. We especially thank Larry Hubert and one of the reviewers for providing us with the MATLAB files for optimal and heuristic least-squares unidimensional scaling methods.

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