Geometric aspects of interior-point algorithms of optimization

建议：DMS 0102628主要研究员：圣母大学数学系Leonid FayBusovich。PI分析与优化问题有关的几何和代数结构。主要目标是进一步扩大内点算法的应用领域。提出了一种基于随机矩阵理论的方法来计算切比雪夫系统生成的锥的特征函数以及这些锥的自协调势垒。这极大地扩展了内点算法的适用范围。特别地，概述了样条线逼近的可能应用。提出了一种构造一大类无限维域自协调势垒的一般方法。我们建议用JB-代数的技巧来继承最近欧几里德-乔丹代数理论在无限维情形下对称规划中令人印象深刻的应用。内点算法理论为分析一大类极其有效的优化算法提供了一个通用的框架。这些算法用于解决在各种约束(例如，时间约束、可用资源等)下寻找最佳可能解的问题。该项目的实现将使我们能够极大地扩展一类可以借助强大的现代计算机来解决的问题。

Proposal: DMS 0102628Principal Investigator: Leonid FayBusovich, Department of Mathematics, University of Notre DameAbstract.PI analyzes geometric and algebraic structures arising in connection with optimization problems. The major goal is to further expand the domain of applications of interior-point algorithms. A method based on the theory of random matrices is proposed to calculate characteristic functions of cones generated by Chebyshev systems and thus self-concordant barriers for these cones. This drastically expands the domain of applicability of interior-point algorithms. In particular, possible applications to spline approximations are outlined. A general approach to the construction of self-concordant barriers for a broad class of infinite-dimensional domains is proposed. The technique of JB-algebras is suggested to carry over the recent impressive applications of the theory of Euclidean Jordan algebras to symmetric programming to the infinite-dimensional situation. Possible control applications are outlined.The theory of interior-point algorithms provides a general framework for the analysis of a broad class of extremely efficient optimization algorithms. These algorithms are used to solve problems aiming at finding the best possible solutions under various constraints (e.g. time constraints , available resources e.t.c ). The realization of the project will enable us to drastically expand a class of problems that can be solved with the help of powerful modern computers.