非线性对称锥规划的同伦算法及应用

项目名称： 非线性对称锥规划的同伦算法及应用

项目编号： No.11301050

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 杨莉

作者单位： 大连理工大学

项目金额： 22万元

中文摘要： 非线性对称锥规划在反馈控制、结构设计、金融等领域有着广泛的应用，并且它包含非线性规划、非线性半定规划等许多经典的数学规划模型，是一个热门的研究领域。目前，大规模非线性半定规划的有效解法和非线性对称锥规划的数值解法的开发仍然是需要进一步研究的课题。本项目拟利用传统的光滑化方法及不精确计算技巧，构造求解具有大规模矩阵不等式约束的非线性半定规划问题的高效实用同伦方法；基于欧氏Jordan代数及非光滑优化理论，构建求解非线性对称锥规划问题的大范围收敛的同伦方法；此外，考虑其特殊模型线性锥规划在风险管理及投资组合中的应用。基于CVaR风险度量，在分布及矩不确定的情况下，构建具有势约束的优化模型，设计有效的求解方法。

中文关键词： 非线性对称锥规划；同伦方法；投资组合优化；条件在险价值；稀疏优化

英文摘要： Nonlinear symmetric cone programming problems have important applications in real world engineering such as feedback control, structural design and finance. Moreover, it contains a wide range of optimization problems such as nonlinear programming, nonlinear semidefinite programming as special cases. The nonlinear symmetric cone programming has become an important research field. The developments of efficient algorithms for solving large sized nonlinear semidefinite programming and numerical methods for solving nonlinear symmetric cone programming still need further research. In this project, by using smoothing methods and inexact methods, we will try to develop efficient homotopy methods for solving nonlinear semidefinite programming problems with large sized matrix inequality constraints. Based on the theory of Euclidean Jordan algebras and nonsmooth optimization, we will also develop globally convergent homotopy methods for solving nonlinear symmetric cone programming problems. Moreover, we will consider to do some researches on applications of linear cone programming to risk management and portfolio optimization problems. Based on the risk control of conditional value at risk, we will consider the cardinality constrained optimization model that describes uncertainty in both the distribution form and moments,

英文关键词： Nonlinear symmetric cone programming；Homotopy method；Portfolio optimization；Conditional value at risk；Sparse optimization