Algorithms for Optimization Problems and Systems of Bilinear Equations over Positive Definite Symmetric Matrices and its Applicatoins

正定对称锥集上的双线性优化问题和双线性方程组的典型例子是半定规划(SDP)和由SDP产生的双线性方程组。在本研究中，我们还研究了作为SDP特例的线性规划(LP)和二阶锥规划(SOCP)。我们过去继续研究并与该项目相关的主题是：(1)SDP的原始-对偶内点算法的多项式分析；(2)SOCP的原始-对偶内点算法的多项式分析；(3)LP的仿射尺度算法全局收敛的反例的构造。在项目期间，我们发表了三篇关于这些主题的论文。我们还研究了以下四个新课题：(4)SDP在图形模型估计中的应用；(5)SOCP在直线动车(新型子弹头列车)磁屏蔽优化设计中的应用；(6)线性规划的Vavasis-Ye多项式时间内点算法的改进，其复杂性仅取决于系数矩阵A；(7)多项式时间原始-对偶内点的推广，算法的影响，使用JB代数来求解尺寸问题。我们还实现了LP的对偶仿射尺度算法和SDP的原始-对偶路径跟踪算法，并尝试用S来解决标准基准问题。

Typical examples of optimization problems and systems of bilinear equations over the set of positive definite symmetic cones are semidefinite programming (SDP) and the systems bilinear equations arising from SDP. In this research, we also studied linear programming (LP) and second-order cone programming (SOCP) as special cases of SDP. The topics we continued to study from the past and related to the project is : (1) Polynomi-ality analysis of primal-dual interior-point algorithms for SDP ; (2) Polynomiality analysis of primal-dual interior-point algorithms for SOCP ; (3) Costruction of a counter example of global convergence of, the affine-scaling algorithm for LP. We published three papers on these topics in the term of the project. There are the following four other new topics we studied in the project : (4) Application of SDP to estimation of graphical models ; (5) Application of SOCP to optimal design of magnetic shielding in linear motor car (new bullet train) ; (6) Modification of the Vavasis-Ye polynomial-time interior-point algorithm for linear programming whose complexity just depends on the coeflicient matrix A ; (7) Extension of polynomial-time primal-dual interior-point, algorithms to inflnite, , dimensional problems using JB algebras. We also implemented the dual affine-scaling algorithm for LP and the primal-dual path-following algorithms for SDP, and made attempt, s to solve standard benchmark problems.