Optimization: Theory, Algorithms, and Applications

[9971852]本文提出了两个研究课题：(1)特征值函数的变分分析(2)非参数总体分析。在第一个课题中，主要研究者将运用非光滑分析和变分几何的现代技术来研究某些谱函数在n × n复值矩阵空间上的变分性质，例如谱横坐标和谱半径。在第二个主题中，首席研究员打算分析和开发非参数版本的人口分析基本问题的算法解决技术。具体来说，主要研究者打算使用最大似然技术来估计与群体变异性相关的潜在概率测量。了解矩阵值映射谱的变分行为对于我们理解和控制离散和连续动力系统是必不可少的。这样的系统在许多实际应用中出现，从能够承受大地震的结构设计到现代飞机的飞行控制。首席研究员的研究计划的结果是为了使在许多工程应用中与光谱变化相关的许多问题的最优性和/或平衡条件的推导首次成为可能。另一方面，人口分析是一种统计方法，用于了解旨在分析与目标人群相关的现象的研究中的主体间变异性。例如，该方法被广泛应用于药代动力学研究，因为它是了解药物在人类和动物中的作用的关键。首席研究员打算分析和开发算法解决技术的非参数版本的这个问题。目标是将这些算法整合到一个软件包中，该软件包目前正在由华盛顿大学的种群动力学资源设施开发。

9971852BurkeTwo topics are proposed for study: (1) the variational analysis of eigenvalue functions and (2) non-parametric population analysis. In the first topic the principal investigator will apply modern techniques of nonsmooth analysis and variational geometry to study the variational properties of certain functions of the spectrum on the space of n x n complex valued matrices, e.g., the spectral abscissa and the spectral radius. In the second topic, the principal investigator intends to analyze and develop algorithmic solution techniques for the non-parametric version of the basic problem of population analysis. Specifically, the principal investigator intends to use maximum likelihood techniques to estimate the underlying probability measure associated with population variability.Understanding the variational behavior of the spectrum of matrix valued mappings is essential to our understanding and control of discrete and continuous dynamical systems. Such systems arise in numerous practical applications ranging from the design of structures that can withstand a major earthquake to flight control for modern aircraft. The results of the principal investigator's research program are intended to make possible for the first time the derivation of optimality and/or equilibrium conditions for numerous problems associated with spectral variations that occur in numerous engineering applications. On the other hand, population analysis is the statistical methodology used to understand inter-subject variability in studies designed to analyze a phenomenon associated with a targeted population. For example, the methodology is widely used in pharmocokinetic studies since it is the key to understanding how drugs behave in humans and animals. The principal investigator intends to analyze and develop algorithmic solution techniques for the non-parametric version of this problem. The goal is to incorporate these algorithms into a software package now being developed by the Resource Facility for Population Kinetics at the University of Washington.