Mathematical programming approaches to three fundamental problems will b described: feature selection clustering and robust representation. The feature selection problem considered is that of discriminating between two sets while recognizing irrelevant and redundant features and suppressing them. This creates a lean model that often generalizes better to new unseen data. Computational results on real data confirm improved generalization of leaner models. Clustering exemplified by the unsupervised learning of patterns and clusters that may exist in a given database and is a useful tool for knowledge discovery in databases (KDD). A mathematical programming formulation of this problem is proposed that is theoretically justifiable and computationally implementable in a finite number of steps. A resulting k-Median Algorithm is utilized to discovery very useful survival curves for breast cancer patients from a medical database. Robust representation is concerned with minimizing trained model degradation when applied to new problems. A novel approach is proposed that purposely tolerates a small error in the training process in order to avoid overfitting data that may contain errors. Examples of applications of these concepts are given.
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