Interior-point methods of optimization: extensions and applications

本文讨论了四个主题：各种半无限规划问题的通用障碍函数的显式和近似计算，应用于多准则和鲁棒控制问题的无限维原始对偶算法，基于凸松弛的无线通信和全局优化，以及内点算法理论在黎曼流形中的推广。为了更好地理解内点算法的数学结构，本文引入了一些新的思想和技术。PI在很大程度上依赖于M. Krein， A. Nudelman， I. Schoenberg关于等周不等式的多维版本和计算新的重要类通用势垒函数的全正矩阵的经典结果，以及用于分析无限维原对偶算法和控制应用的无限维Jordan代数理论。PI为全局优化带来了一些与希尔伯特恒等式相关的数论思想和所谓的球形设计的经典构造。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、家庭土地安全中的识别应用）、环境政策、贝叶斯统计中的鲁棒性、回归中的最优实验设计、工业过程的效率、电气工程中的过滤设计等。解决这类问题的算法问题的进展可能会对各种应用程序产生巨大的影响（特别是在需要快速和最佳在线决策的情况下）。开发用于解决鲁棒、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、股票市场行为预测和航天器控制等各种应用中可能很重要。

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.