解结构型DC规划问题的数值算法及其应用研究

Structured DC programs have wide applications in the fields of communication engineering, statistics and so on. Because of the special structure as well as the nonconvexity of structured DC programs, their directional stationary points have attracted much attention. Directional stationary points are a class of strongly stationary points. Most existing numerical algorithms can only compute weakly stationary points. In order to meet the needs of theories and practical applications, some numerical algorithms were proposed to solve directional stationary points of structured DC programs. The aim of this project is to propose numerical algorithms for computing the directional stationary point of structured DC programs: 1、we shall present successive convex approximation methods to solve the directional stationary point of structured DC programs and analyze their convergence under proper conditions; 2、we shall propose a l_1 penalty method to solve structured DC programs, and prove that under pointwise Slater constraint qualifications every accumulation point of the sequence of iterates generated by the l_1 penalty method is a B-stationary point; 3、we also present a l_2 penalty method and an augmented Lagrangian method to solve structured DC programs; 4、we shall construct DC models and DCA algorithms to solve structured optimization problems and piecewise linear systems.

结构型DC规划问题在通信工程、统计学等领域有广泛的应用。由于问题具有的特殊结构和非凸性质，它们的方向稳定点(如D-稳定点和B-稳定点)受到了密切关注。方向稳定点是一类强稳定点。现有的绝大部分数值算法只能求解问题的弱稳定点。为了满足理论研究和实际应用的需要，最近，一些学者提出了数值算法来解结构型DC规划问题的方向稳定点。本项目拟继续研究数值算法求解结构型DC规划问题的方向稳定点：1、拟构造连续凸逼近算法解结构型DC规划问题, 在适当的条件下分析算法的收敛性；2、拟构造l_1罚方法解结构型DC规划问题，并且证明在逐点形式的Slater约束品性下算法生成的序列的聚点为结构型DC规划问题的B-稳定点；3、拟构造l_2罚方法和增广拉格朗日法解结构型DC规划问题，并且分析算法的收敛性；4、拟构造DC模型和DCA算法解特殊结构的优化问题和分片线性方程组。

结构型DC规划在信息科学和统计学等领域有广泛的应用背景，研究算法求解结构型DC规划的强稳定点受到了国内外学者的广泛关注。本项目提出了新的算法解结构型DC规划和离散线性互补系统等。具体研究内容如下：第一、提出了罚方法和增广拉格朗日法解DC约束DC规划，在逐点Slater约束品性条件下，证明了两类算法产生的迭代序列的极限点是原问题的B-稳定点；第二、提出了外推的增强邻近DC算法解无约束的结构型DC规划，证明了算法产生的迭代序列的每一个聚点是原问题的D-稳定点并且在适当的条件下研究了序列的收敛性；第三、提出了新的一阶迭代算法解一类无约束结构型优化问题，在一定条件下证明了算法产生的迭代序列的聚点是优化问题的一阶稳定点；此外，还提出了广义牛顿法解一类离散线性互补系统，在适当的条件下证明了算法具有全局收敛性并且经过有限步迭代就能求得原问题的解。