Applied Mathematical Programming

这个项目是由最近数学编程在机器学习、数据挖掘、医学诊断和预后方面的应用驱动的，包括以下问题：特征选择：这是通过使用尽可能少的特征的分离平面来区分n维特征空间中的两个有限集的问题。用非负实线上的凹指数逼近线性约束数学规划重构目标中的阶跃函数。这导致了一种非常快速的迭代线性规划算法，它在有限的步骤中终止。扩展到线性系统的最小支持解、互补问题和具有最小隐藏单元数的神经网络。聚类：在n空间中确定k个中心的聚类问题，使m个给定点到其最近中心的距离之和最小。为了将问题表述为多面体集合上分段线性凹函数的最小化问题，使用了1范数距离。有限连续线性近似算法很快在一个固定点终止，对于乳腺癌医学数据库来说，这是一个非常有用的挖掘不同生存曲线的工具。将扩展到围绕其他几何实体（如平面或凸集）的聚类，具有特征选择的聚类以及在医学数据库中知识发现的应用。健壮的表示：数据库中的关系将以在数据库适度变化时仍然有效的方式建模。一个简单的线性模型表明，如果在构建模型时故意容忍一个足够小的误差，那么对于一类广泛的扰动，该模型将比传统的零误差容忍得到的模型更准确。特征选择在容忍训练中的作用，以及它在神经网络泛化中的应用将被研究。

This project is driven by recent applications of mathematical programming to machine learning, data mining, medical diagnosis and prognosis and include the following problems: FEATURE SELECTION: This is the problem of discriminating between two finite sets in n-dimensional feature space by a separating plane that utilizes as few of the features as possible. A step function that appears in the objective of a linearly-constrained mathematical programming reformulation is approximated by a concave exponential on the nonnegative real line. This leads to a very fast iterative linear-programming-based algorithm that terminates in a finite number of steps. Extensions to minimum- support solutions of linear systems and complementarity problems and neural networks with minimal number of hidden units will be investigated. CLUSTERING: The clustering problem of determining k centers in n- space such that the sum of distances of each of m given points to its nearest center is minimized. A 1-norm distance is used in order to formulate the problem as that of minimizing a piecewise- linear concave function on a polyhedral set. A finite successive linear approximation algorithm terminates very quickly at a stationary point which, for breast cancer medical databases, is extremely useful as a mining tool for distinct survival curves. Extensions to clustering around other geometric entities such as planes or convex sets, clustering with feature selection and applications to knowledge discovery in medical databases will be pursued. ROBUST REPRESENTATION: Relations within a database will be modeled in a manner that remains valid when the database changes moderately. A simple linear model shows that if a sufficiently small error is PURPOSELY TOLERATED in constructing the model, then for a broad class of perturbations the model will be a more accurate than one obtained by a conventional zero error tolerance. The effect of feature selection within tolerant training, as well its use to improve generalization in neural networks will be studied.