Applications, Algorithms and Theory of Mathematical Programming

非常有希望的数学规划技术应用于医疗诊断和预测以及一般的机器学习。利用威斯康星大学医院目前使用的高精度乳腺癌诊断系统的五年经验，开发了一个基于线性规划的预后系统。通过在并行处理机之间分配问题的成分(约束、梯度或/和变量)，开发了求解大规模约束优化问题的并行算法。每个加工商都完全有责任改变自己的问题成分，同时允许其他成分以有限的方式改变。处理器共享计算出的新信息，然后执行快速同步并重复该过程。初步的算法原型已经在公开可用的测试问题和重要的现实应用上得到了成功的测试，并具有较高的并行化效率。对于可能不一致的不等式组、规划和互补问题，给出了误差界。这里的新奇思想是，系统可能是不可解的，并且无论系统是否可解，界都是有意义的。在后一种情况下，它限定了所考虑的点与系统的最小误差解集合之间的距离。研究了一些机器学习问题的新的数学规划方法。特别是，一种新的二次规划模型正在研究中，它适用于神经网络中的级联结构，以及通过分离平面最小化误分类点数这一固有困难问题的另一个模型。

Highly promising mathematical programming techniques applied to medical diagnosis and prognosis as well as to machine learning in general. Five years of experience with a highly accurate system for breast cancer diagnosis, in current use at University of Wisconsin Hospitals, is drawn upon to develop a linear-programming-based prognostic system. Parallel algorithms for the solution of large-scale constrained optimization problems are developed, by distributing ingredients of the problem (constraints, gradients, or/and variables) among parallel processors. Each processor has complete responsibility for varying its own problem ingredients, while allowing the other ingredients to vary in a restricted fashion. The processors share new information computed, then a fast synchronization is performed and the process is repeated. Preliminary algorithm prototypes have been tested successfully with some high parallelization efficiency on publicly available test problems and significant real-world applications. Error bounds are developed for possibly inconsistent systems of inequalities, programs and complementarity problems. The novel idea here is that the system may be unsolvable, and the bounds are meaningful whether the system is solvable or not. In the latter case, it bounds the distance between the point under consideration, and the set of least error solutions of the system. New mathematical programming approaches to some machine learning problems are studied. In particular a new quadratic programming model for a cascade architecture in neural networks is being studied, as is another one for the inherently difficult problem of minimizing the number of misclassified points by a separating plane.