INTERIOR-POlNT METHODS FOR OPTIMIZATION PROBLEMS IN SOCIAL SYSTEMS

我们对求解社会系统最优化问题的内点方法进行了基础性研究。在这一年里，我们的研究主要致力于发展线性规划和半定规划的内点方法，并致力于分析这些方法的全局收敛和局部收敛。线性规划问题是最基本的数学规划问题。我们对线性规划的内点方法进行了两个重要的研究。首先，为了加快内点法的局部收敛速度，我们研究了一种高度跟踪中心路径的算法。作为研究的结果，我们提出了一种高阶不可行内点算法。其次，我们感兴趣的是获得一个初始内点以执行算法的问题，为此，我们研究了两个具有平凡初始点的线性规划的自对偶系统，我们不仅研究了线性规划，还研究了凸规划、半定规划和半无限规划。在求解半定规划问题的内点方法中，有许多搜索方向。我们研究了这样一个方向的自对偶子族。我们还研究了求解线性和二次半无限规划问题的内点法，并提出了一个凸半无限规划的对偶参数化算法。

We have performed a basic research on the interior point methods for solving optimization problems in social systems. In this year, our research mainly devoted to the development of interior point methods for linear programming and semidefinite programming, and devoted to the analysis of the global convergence and the local convergence of these methods.Most of the optimization problems iii social systems could be modeled as mathematical programming problems. The linear programming problem is the most fundamental mathematical programming problem. We performed two important research on the interior-point methods for linear programming. Firstly, in order to speed up the local convergence of interior point methods, we investigated an algorithm which trace the central path in high degree. As a result of this research, we proposed a high order infeasible interior point algorithm. Secondly, we are interested in the problem to get an initial interior point to perform an algorithm, For this purpose, we investigated two self-dual systems for linear programming, which have trivial initial points.We studied not only linear programming, but also convex programming, semidefinite programming, and semi-infinite programming. There are many search directions in interior-point methods for solving semidefinite programming problems. We investigated a self-dual subfamily of such directions. We also studied interior-point methods for solving linear and quadratic semi-infinite programming problems and proposed a dual-parameterization algorithm for convex semi-infinite programming.