Global Optimization in Any Minkowski Metric: A Permutation-Translation Simulated Annealing Algorithm for Multidimensional Scaling

It is well known that considering a non-Euclidean Minkowski metric in Multidimensional Scaling, either for the distance model or for the loss function, increases the computational problem of local minima considerably. In this paper, we propose an algorithm in which both the loss function and the composition rule can be considered in any Minkowski metric, using a multivariate randomly alternating Simulated Annealing procedure with permutation and translation phases. The algorithm has been implemented in Fortran and tested over classical and simulated data matrices with sizes up to 200 objects. A study has been carried out with some of the common loss functions to determine the most suitable values for the main parameters. The experimental results confirm the theoretical expectation that Simulated Annealing is a suitable strategy to deal by itself with the optimization problems in Multidimensional Scaling, in particular for City-Block, Euclidean and Infinity metrics.

Authors and Affiliations

1. University of Granada, Granada, Spain

   J. Fernando Vera

2. Department of Psychology, Faculty of Social and Behavioural Sciences, Leiden University, The Netherlands

   Willem J. Heiser

3. University of Costa Rica, Turrialba, Costa Rica

   Alex Murillo

Corresponding authors

Correspondence to J. Fernando Vera, Willem J. Heiser or Alex Murillo.

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