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D.C. Programming Approach to the Multidimensional Scaling Problem

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From Local to Global Optimization

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 53))

Abstract

The paper is devoted to the solution of the metric Euclidean Multidimensional Scaling Problem (MDS) and the Euclidean distance geometry problem by methods of d.c. programming developed earlier by the authors. The algorithms, called DCA (DC Algorithms), are based on the duality theory and local optimality conditions for d.c. programming. Different regularization techniques are used in order to improve the qualities (robustness, stability, convergence rate and globality of computed solutions) of DCA. Lagrangian duality without gap allows us to find interesting equivalent forms of (MDS) and a very simple expression of the dual objective function which can be used for checking globality of solutions computed by DCA. DCA (with and without regularization techniques) is described for solving both MDS problems. Requiring only matrix-vector products and only one Cholesky factorization, it is quite simple, and allows exploiting sparsity in large-scale problems. Moreover the well-known majorization method by de Leeuw can be viewed as a special case of DCA. DCA (which globally solves the trust region problem) is also presented in its parametric version to deal with the Euclidean metric MDS problem. Finally extensive numerical experiments are reported which show the robustness and efficiency of DCA for solving the Euclidean metric MDS problem, especially in the largescale setting. They also show the globality of computed solutions in the case where the dissimilarities are the Euclidean distances between the objects.

This work was partially supported by the computer resources financed by Contrat de Plan Interregional du Bassin Parisien-Pole Interregional de Modelisation en Sciences pour Ingenieurs

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Authors and Affiliations

  1. Mathematical Modelling and Applied Optimization Group, Laboratory of Mathematics — CNRS UPRES-A 60 85, National Institute for Applied Sciences-Rouen, BP 8, F 76 131, Mont Saint Aignan Cedex, France

    Le Thi Hoai An & Pham Dinh Tao

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  1. Le Thi Hoai An
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  2. Pham Dinh Tao
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Editors and Affiliations

  1. Technical University of Crete, Chania, Crete, Greece

    Athanasios Migdalas

  2. University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  3. Linköping University, Norrköping, Sweden

    Peter Värbrand

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An, L.T.H., Tao, P.D. (2001). D.C. Programming Approach to the Multidimensional Scaling Problem. In: Migdalas, A., Pardalos, P.M., Värbrand, P. (eds) From Local to Global Optimization. Nonconvex Optimization and Its Applications, vol 53. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5284-7_11

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  • DOI: https://doi.org/10.1007/978-1-4757-5284-7_11

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