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

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

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Article ID: iaor20126636
Volume: 54
Issue: 4
Start Page Number: 689
End Page Number: 706
Publication Date: Dec 2012
Journal: Journal of Global Optimization
Authors: , ,
Keywords: programming: convex, matrices
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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