Geometric aspects of interior-point algorithms of optimization

提案：DMS 0102628首席研究员：Leonid FayBusovich，圣母大学数学系PI分析与最优化问题有关的几何和代数结构。主要目标是进一步拓展内点算法的应用领域。提出了一种基于随机矩阵理论的计算切比雪夫系统生成的锥体特征函数的方法，从而计算出这些锥体的自调和障碍。这极大地扩展了内点算法的适用范围。特别地，概述了样条近似的可能应用。提出了一类广泛的无限维域的自和谐屏障构造的一般方法。建议使用jb -代数技术将欧几里得约当代数理论在无限维情况下的对称规划中的令人印象深刻的应用延续下去。概述了可能的控制应用。内点算法理论为分析一类非常高效的优化算法提供了一个总体框架。这些算法用于解决在各种约束条件下（如时间限制、可用资源等）寻找最佳可能解决方案的问题。这个项目的实现将使我们能够在强大的现代计算机的帮助下大大扩展一类问题。

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.