Applied Mathematical Programming

该项目是由最近的应用数学规划机器学习，数据挖掘，医疗诊断和预后，包括以下问题： 功能选择：这是在n维特征空间中通过利用尽可能少的特征的分离平面来区分两个有限集的问题。线性约束数学规划的目标函数中出现的阶跃函数，用非负真实的直线上的凹指数逼近。 这导致了一个非常快速的迭代线性规划为基础的算法，终止于有限数量的步骤。将研究线性系统和互补问题的最小支撑解的扩展以及具有最小隐单元数的神经网络。 聚类：在n-空间中确定k个中心，使m个给定点到最近中心的距离之和最小的聚类问题。 利用1-范数距离将问题转化为多面体集上的分段线性凹函数极小化问题。有限连续线性逼近算法在稳定点处非常快地终止，对于乳腺癌医疗数据库来说，这作为不同生存曲线的挖掘工具非常有用。扩展到其他几何实体，如平面或凸集，聚类与特征选择和应用程序的知识发现在医学数据库中的聚类将继续进行。 健壮的表示：数据库中的关系将以一种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，如果一个足够小的误差是非常宽容的构建模型，那么对于一个广泛的扰动类的模型将是一个更准确的比一个传统的零误差容限。 将研究容差训练中特征选择的效果，以及其用于提高神经网络的泛化能力。

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.