Branch and bound algorithm for multidimensional scaling with city-block metric

A two level global optimization algorithm for multidimensional scaling (MDS) with city-block metric is proposed. The piecewise quadratic structure of the objective function is employed. At the upper level a combinatorial global optimization problem is solved by means of branch and bound method, where an objective function is defined as the minimum of a quadratic programming problem. The later is solved at the lower level by a standard quadratic programming algorithm. The proposed algorithm has been applied for auxiliary and practical problems whose global optimization counterpart was of dimensionality up to 24.

[1]  Jan de Leeuw,et al.  Differentiability of Kruskal's stress at a local minimum , 1984 .

[2]  Antanas Zilinskas,et al.  Two level minimization in multidimensional scaling , 2007, J. Glob. Optim..

[3]  Phipps Arabie,et al.  Was euclid an unnecessarily sophisticated psychologist? , 1991 .

[4]  Paul E. Green,et al.  Multidimensional Scaling: Concepts and Applications , 1989 .

[5]  Michael J. Brusco.  A Simulated Annealing Heuristic for Unidimensional and Multidimensional (City-Block) Scaling of Symmetric Proximity Matrices , 2001, J. Classif..

[6]  D. Defays.  A short note on a method of seriation , 1978 .

[7]  Henri Xhaard,et al.  Conserved structural, pharmacological and functional properties among the three human and five zebrafish α2‐adrenoceptors , 2005, British journal of pharmacology.

[8]  L. Hubert,et al.  Multidimensional scaling in the city-block metric: A combinatorial approach , 1992 .

[9]  Willem J. Heiser,et al.  City-Block Scaling: Smoothing Strategies for Avoiding Local Minima , 1998 .

[10]  Michael J. Brusco,et al.  Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming , 2005 .

[11]  M. Brusco,et al.  Branch-and-Bound Applications in Combinatorial Data Analysis , 2005 .

[12]  Antanas Zilinskas,et al.  A hybrid method for multidimensional scaling using city-block distances , 2008, Math. Methods Oper. Res..

[13]  P. Groenen,et al.  Modern multidimensional scaling , 1996 .

[14]  R. Graham,et al.  Identification of Critical Determinants of α1-Adrenergic Receptor Subtype Selective Agonist Binding (*) , 1995, The Journal of Biological Chemistry.

[15]  A. ilinskas,et al.  Parallel hybrid algorithm for global optimization of problems occurring in MDS-based visualization , 2006 .

[16]  P. Groenen,et al.  The majorization approach to multidimensional scaling for Minkowski distances , 1995 .

[17]  J. Wikberg,et al.  [3H]RS79948-197 binding to human, rat, guinea pig and pig alpha2A-, alpha2B- and alpha2C-adrenoceptors. Comparison with MK912, RX821002, rauwolscine and yohimbine. , 1998, European journal of pharmacology.

[18]  P. Groenen,et al.  Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing , 1999 .

[19]  Julius Žilinskas.  MULTIDIMENSIONAL SCALING IN PROTEIN AND PHARMACOLOGICAL SCIENCES , 2006 .