Mathematical Sciences: Algorithms for Convex Programming-Interior Point and Proximal Point Methods

该项目将支持大规模凸规划问题的算法分析和开发的研究。最通用的两种算法是内点法和基于近似点算法的分裂算法。具体的研究领域将包括：(1)内点变换，包括分析内点方法中的映射，即将内部可行对发送到其坐标乘积：(2)不可行内点方法，包括弥合可行内点方法和不可行内点方法之间的差距；(3)邻近点方法的收敛，包括对邻近点方法及其修正的全局和局部收敛行为的分析；(4)非线性问题的内点方法，涉及对特殊类型的非线性规划和单调互补问题的实用内点方法，特别是原始-对偶方法的发展。求解大规模凸规划问题的两种最重要的现代算法是内点法和邻近点法。这两种方法都对现代数值求解技术产生了重大影响，这些技术以前被认为是难以解决的问题，因为它们的规模巨大。

This project will support research in the analysis and development of algorithms for large-scale convex programming problems. Two of the most versatile algorithms have been interior point methods and splitting methods based on the proximal point algorithm. Specific areas of research will include: (1) the interior point transformation, involving the analysis of the mapping in interior point methods that sends an interior feasible pair to its coordinatewise product; (2) infeasible interior point methods, including bridging the gap between the feasible and infeasible interior point methods; (3) convergence of the proximal point method, including the analysis of the global and local convergence behavior of the proximal point method and its modifications; and (4) interior point methods for nonlinear problems, involving the development of practical interior point methods, especially primal-dual methods, for special classes of nonlinear programs and monotone complementarity problems. Two of the most important modern algorithms for the numerical solution of large-scale convex programming problems have been interior point methods and proximal point methods. Both of these methods have had a major impact on modern numerical solution techniques for problems that were previously thought to be intractable because of their massive size.