Interior-point methods of optimization: extensions and applications

内点优化方法：扩展和应用

• 批准号: 0402740
• 负责人: Leonid Faybusovich
• 金额: $ 18万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0402740&HistoricalAwards=false
• 关键词: Interior; point; methods; optimization; extensions

Four topics are discussed in the proposal: explicit and approximate computations of universal barrier functions for various classes of semi-infinite programming problems, infinite-dimensional primal-dual algorithms with applications to multi-criteria and robust control problems, wireless communication and global optimization based on convex relaxations and a generalization of the theory of interior-point algorithms to Riemannian manifolds. Proposal introduces several new ideas and techniques for better undersanding the mathematical structure of interior-point algorithms. PI heavily relies upon classical results of M. Krein, A. Nudelman, I. Schoenberg on multidimensional versions of isoperimetric inequalities and totally positive matrices in computation of new important classes of universal barrier functions, on the theory of infinite-dimensional Jordan algebras for the analysis of infinite-dimensional primal-dual algorithms and control applications. PI brings some number-theoretic ideas related to Hilbert identities and classical constructions of so-called spherical designs to the global optimization. Semi-infinite programming problems appear in various applications: separation of sets in pattern recognition (e.g. medical diagnostics , identification with applications in home land security), environmental policies, robustness in Bayesian statistics, optimal experimental design in regression, the efficiency of industrial processes, filtering design in electrical engineering etc. The progress in resolving algorithmic issues for this class of problems may potentially have a tremendous impact on various applications (especially, in the situations where fast and optimal online decisions are necessary). Development of software for solving robust , multi-criteria control problems may be of importance in such diverse applications as stabilization of various complex structures in extreme situations (earthquakes, overloads of energy systems), prediction of the behavior of stock markets and control of spacecrafts.

四个主题的建议进行了讨论：明确和近似计算的通用障碍函数的各类半无限规划问题，无限维原始对偶算法与应用程序的多准则和鲁棒控制问题，无线通信和全局优化的基础上凸松弛和推广的理论的边界点算法黎曼流形。提出了一些新的思想和技术，以更好地理解的数学结构的邻域点算法。PI严重依赖于M. Krein，A.努德曼岛勋伯格在多维版本的等周不等式和完全正矩阵的计算新的重要类的普遍障碍函数，对理论的无限维约旦代数的分析无限维原始对偶算法和控制的应用。PI带来了一些数论思想有关希尔伯特恒等式和经典结构的所谓的球形设计的全局优化。 半无限规划问题出现在各种应用中：模式识别中的集合分离（例如医疗诊断、在国土安全中的应用）、环境政策、贝叶斯统计的稳健性、回归中的最佳实验设计、工业过程的效率、解决这类问题的算法问题的进展可能对各种应用产生巨大影响（特别是在需要快速和最佳在线决策的情况下）。开发解决鲁棒多准则控制问题的软件在极端情况下（地震、能源系统过载）各种复杂结构的稳定、股票市场行为的预测和航天器的控制等各种应用中可能具有重要意义。