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salesman problem. he approach borrows its philosophy from theoretical physics.
By analogy the system is raised to a high temperature in which the individuals move
freely with respect to each other. As cooling occurs this mobility is lost. Some align-
ment occursbetween the individuals, and the system approaches a minimum energy
state. To try to achieve the desired global minimum, the cooling must occur slowly.
hisprocedurehasbeenencapsulatedwithinpublishedalgorithms(Goffeetal., ;
Corana et al., ). he approach differs from the conventional gradient descent
method in that an occasional step away from an apparent minimum might be used
to assist in escaping from local minima.
hemajorizationapproachtominimizationwasfirstproposedbydeLeeuw( )
for use with MDS. Essentially, a complicated function f
(
x
)
is replaced by a more
manageableauxiliaryfunction g
(
x, y
)
suchthatforeach x inthedomainof f , f
(
x
)
.he
function g is called the majorizing function. An initial value x is used and then
g
g
(
x, y
)
,foraparticulary in the domain of g, and also so that f
(
y
)=
g
(
y, y
)
is minimized with respect to x.Letthevalueofx, which gives rise to the
minimum, be x .heng
(
x, x
)
(
x, x
)
is minimized with respect to x, and so on until con-
vergence.
References
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: -
Anderberg, M.R. ( ). Cluster Analysis for Applications. Academic, New York
Baulieu, F.B. ( ). J Classificat : -
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York
BP ( ). http://www.bp.com
Carroll, J.D. and Chang, J.J. ( ) Analysis of individual differences in multidimen-
sional scaling via an n-way generalization of “Eckart-Young” decomposition. Psy-
chometrika : -
Charbonneau, P. ( ). Astrophys J Suppl Ser : -
Charbonneau, P. and Knapp, B. ( ). http://downloadhaoucaredu/archive/pikaia/
Corana, A., Marchesi, M., Martini, C. and Ridella, S. ( ). ACM Trans Math Sotw
: -
Cormack, R.M. ( ). JRStatSocA : -
Cox,T.F.( ).An Introduction to Multivariate Data Analysis.HodderArnold,Lon-
don
Cox, T.F. and Cox, M.AA. ( ). Multidimensional Scaling. Chapman & Hall/CRC,
BocaRaton, FL
de Leeuw, J. ( ). Applications of convex analysis to multidimensional scaling. In:
Barra,J.R.,Brodeau,F.,Romier,G.,vanCutsen,B.(eds)Recent Developments in
Statistics. North Holland, Amsterdam
Diday,E.andSimon,J.C.( ).Clustering analysis. In:Fu,K.S.(ed) Communication
and Cybernetics Digital Pattern Recognition. Springer, Berlin Heidelberg New
York
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