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the objective function of Problem 2 is a constant and can be dropped, though for
the sake of completeness we will retain the term throughout the subsequent formu-
lations in the paper. Problems 1 and 2 are Mixed Integer Nonlinear Programming
(MINLP) problems with bilinear terms in the objective function and the first set
of constraints. To handle the nonlinearities formed by the product of variables
w
ij
and
z
jk
,newvariables
y
ijk
along with additional constraints [19] are defined as
follows:
y
ijk
=
w
ij
z
jk
(16.1)
z
jk
(1
z
jk
(1
z
jk
−
−
w
ij
)
≤
y
ijk
≤
z
jk
−
−
w
ij
)
(16.2)
z
jk
w
ij
≤
z
jk
w
ij
,
y
ijk
≤
∀
i,
∀
j,
∀
k
(16.3)
The introduction of
y
ijk
and the additional constraints reduces the formula-
tion to an equivalent Mixed-Integer Linear Programming (MILP) problem, but
results in an inordinately large number of variables. Thus, there is a need for new
approaches to address large datasets.
16.2.2.3.
The GOS Algorithm for Clustering
The introduction of the bilinear variable
y
ijk
results in a large number of vari-
ables to be considered. In a problem with over 2000 data points, each having 24
features, to be placed into over 380 clusters, the number of variables to be consid-
ered numbers over 18 million. Without introducing the
y
ijk
variables will leave
the problem in a nonlinear form. Mixed-integer nonlinear programming (MINLP)
problems are considered extremely difficult. Theoretical advances and prominent
algorithms for solving MINLP problems are addressed in [19, 20, 22].
The general form of a MINLP problem is:
min
C
(
x,y
)
(Problem 3)
s.t.
h
(
x,y
)=0
g
(
x,y
)
≤
0
(0
,
1)
m
,x
n
y
∈
∈
Here, x represents the continuous variables in real space and y, the integer
variables. For simplicity here, y is assumed to be binary. In addition, C(x,y) is
the objective function, h(x,y) represents the set of equality constraints, and g(x,y)
is the set of inequality constraints. We propose here a variant of the Generalized
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