Journal of Global Optimization 23: 63–80, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
63
A method of truncated codifferential with
application to some problems of cluster analysis
V.F. DEMYANOV
1
, A.M. BAGIROV
2
and A.M. RUBINOV
2
1
Department of Applied Mathematics, Saint-Petersburg State University, Russia;
2
School of
Information Technology and Mathematical Sciences, University of Ballarat, Australia (e-mail:
amr@ballarat.edu.au)
Abstract. A method of truncated codifferential descent for minimizing continuously codifferentiable
functions is suggested. The convergence of the method is studied. Results of numerical experiments
are presented. Application of the suggested method for the solution of some problems of cluster
analysis are discussed. In numerical experiments Wisconsin Diagnostic Breast Cancer database was
used.
Key words: subdifferential, quasidifferential, codifferential, truncated codifferential, cluster ana-
lysis.
1. Introduction
A mathematical formalization of one of the main problems of cluster analysis leads
to the following global optimization problem: for a given set of points a
i
∈ R
n
,i =
1,...,m find a collection ¯ x = ( ¯ x
1
,..., ¯ x
p
) of pn-dimensional vectors, which is
a solution of the following problem:
f(x
1
,...,x
p
) =
m
i =1
min
l =1,...,p
‖x
l
- a
i
‖ -→ min subject to
x
l
∈ S ⊂ R
n
,l = 1,...,p, (1)
where S is a compact convex set in R
n
. As a rule m is a large number and p is
substantially less than m.
The objective function in (1) is a DC function (see, for example, Tuy, 1998),
that is, f can be represented as the difference of two convex functions. In fact
f(x) =
m
i =1
p
l =1
‖x
l
- a
i
‖-
m
i =1
max
r
l =r
‖x
l
- a
i
‖. (2)
Since f is a DC function it follows that f is continuously codifferentiable. (For
definition and properties of codifferentiable functions see Demyanov and Rubinov
(1995) and also Section 2 below.) There are many continuous codifferentials for
the function f at a point x . Starting from the functions x
l
→‖x
l
- a
i
‖ and