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On some convexity properties of the Least Squares Method for pairwise comparisons matrices without the reciprocity condition

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Abstract

The relaxation of the reciprocity condition for pairwise comparisons is revisited from the optimization point of view. We show that some special but not extreme cases of the Least Squares Method are easy to solve as convex optimization problems after suitable nonlinear change of variables. We also give some other, less restrictive conditions under which the convexity of a modified problem can be assured, and the global optimal solution of the original problem found by using local search methods. Mathematical and psychological justifications for the relaxation of the reciprocity condition as well as numerical examples are provided.

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Authors and Affiliations

  1. Research Group of Operations Research and Decision Systems, Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, 1518, Budapest, Hungary

    J. Fülöp

  2. Computer Science, Laurentian University, Sudbury, ON, P3E 2C6, Canada

    W. W. Koczkodaj

  3. Department of Mathematics, Case Western Reserve University, Cleveland, OH, 44106-7058, USA

    S. J. Szarek

  4. Université Pierre et Marie Curie-Paris 6, 75252, Paris, France

    S. J. Szarek

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  1. J. Fülöp
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  2. W. W. Koczkodaj
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  3. S. J. Szarek
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Correspondence to J. Fülöp.

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Fülöp, J., Koczkodaj, W.W. & Szarek, S.J. On some convexity properties of the Least Squares Method for pairwise comparisons matrices without the reciprocity condition. J Glob Optim 54, 689–706 (2012). https://doi.org/10.1007/s10898-011-9785-z

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  • DOI: https://doi.org/10.1007/s10898-011-9785-z

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