Integer Programming Problem Formulation

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ILP Problem Formulation
Ajay Kr. Dhamija (N-1/MBA PT 2006-09)

Integer linear programming is a very important class
of problems, both algorithmically and combinatori-
ally.Following are some of the problems in computer
Science ,relevant to DRDO, where integer linear Pro-
gramming can be e®ectively used to ¯nd optimum so-
1. Pattern Classi¯cation
2. Multi Class Data Classi¯cation
3. Image Contrast Enhancement
Pattern Classi¯cation is being extensively used for
automatic speech recognition, classi¯cation of text
into several categories (e.g. spam/non-spam email
messages), the automatic recognition of handwritten
words, or the automatic recognition of images of
human faces .I present here ,a minimum sphere cov-
ering approach to pattern classi¯cation that seeks to
construct a minimum number of spheres to represent
the training data and formulate it as an integer
programming problem. Using soft threshold functions,
we can further derive a linear programming problem
whose solution gives rise to radial basis function
(RBF) classi¯ers and sigmoid function classi¯ers. In
contrast to traditional RBF and sigmoid function
networks, in which the number of units is speci-
¯ed a priori, this method provides a new way to
construct RBF and sigmoid function networks that
explicitly minimizes the number of base units in the
resulting classi¯ers. This approach is advantageous
compared to SVMs with Gaussian kernels in that
it provides a natural construction of kernel matrices
and it directly minimizes the number of basis functions.
Traditional approaches for data classi¯cation ,
that are based on partitioning the data sets into two
groups, perform poorly for multi-class data classi¯ca-
tion problems. The proposed approach is based on
the use of hyper-boxes for de¯ning boundaries of the
classes that include all or some of the points in that
set. A mixed-integer programming model is developed
¤Computer Scientist, Defence R&D Org., Min of
Defence, Delhi-110054.,, a k Home- k dhamija/
for representing existence of hyper-boxes and their
boundaries. In addition, the relationships among
the discrete decisions in the model are represented
using propositional logic and then converted to their
equivalent integer constraints using Boolean algebra.
Image Contrast Enhancement and Image Recon-
struction are being used for extracting knowledge from
satellite images of the battle¯eld or other terrains.This
method has already been described in LP problem
formulation in I semester assignment.
Keywords: Integer linear Programming ,Pattern
Classi¯cation ,Multi Class data classi¯cation , Image
Reconstruction ,radial basis function (RBF) classi¯ers
, sigmoid function , SVM , Kernel and propositional
1 Pattern Classification Via Integer linear Programming
Given the space in which objects to be classi¯ed are
represented, a classi¯er partitions the space into dis-
joint regions and associates them with di®erent classes.
If the underlying distribution is known, an optimal
partition of the space can be obtained according to
the Bayes decision rule. In practice, however, the
underlying distribution is rarely known, and a learning
algorithm has to generate a partition that is close to
the optimal partition from the training data. The RCE
network (1) is a learning algorithm that constructs a
set of regions, e.g., spheres, to represent each pattern
class. It is easy to see that, with only a few spheres,
there is a great chance that the training error will be
high. With an excessively large number of spheres,
however, the training error can be reduced, but at
the expense of over¯tting the data and degrading the
performance on future data. Similar problems also
exist in the radial basis function (RBF) networks and
multi-layer sigmoid function networks. Therefore, a
good learning...
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