# Integer Programming Problem Formulation

**Topics:**Linear programming, Optimization, Operations research

**Pages:**16 (4120 words)

**Published:**August 12, 2008

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

Abstract

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-

lutions.

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. email:akdhamija@dipr.drdo.in,

dhamija.ak@gmail.com, a k dhamija@yahoo.com. Home-

page:www.geocities.com/a 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

logic

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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