ILP Problem Formulation 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 effectively used to find optimum so- lutions. 1. Pattern Classification 2. Multi Class Data Classification 3. Image Contrast Enhancement Pattern Classification is being extensively used for automatic speech recognition, classification 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 classification 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) classifiers and sigmoid function classifiers. In contrast to traditional RBF and sigmoid function networks, in which the number of units is speci- fied 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 classifiers. 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 classification , that are based on partitioning the data sets into two groups, perform poorly for multi-class data classifica- tion problems. The proposed approach is based on the use of hyper-boxes for defining 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 battlefield or other terrains.This method has already been described in LP problem formulation in I semester assignment. Keywords: Integer linear Programming ,Pattern Classification ,Multi Class data classification , Image Reconstruction ,radial basis function (RBF) classifiers , sigmoid function , SVM , Kernel and propositional logic 1 Pattern Classification Via Integer linear Program- ming Given the space in which objects to be classified are represented, a classifier partitions the space into dis- joint regions and associates them with different 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 overfitting 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 algorithm has to strike a delicate balance between the training error and the complexity of the model.