非线性对称锥规划的内点算法及在最优控制中的应用

项目名称： 非线性对称锥规划的内点算法及在最优控制中的应用

项目编号： No.11471211

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 王国强

作者单位： 上海工程技术大学

项目金额： 68万元

中文摘要： 非线性对称锥规划（NSCP）是指对称锥上的非线性规划问题，是当前国际最优化领域的前沿和热点问题，在最优控制、经济管理、工程技术等领域有广泛应用。国际上线性对称锥规划的内点算法研究非常成功，但NSCP的内点算法研究比较欠缺。本项目旨在系统研究NSCP的内点算法及在最优控制中的应用。理论上，利用欧几里德若当代数和T-代数建立NSCP的对偶理论及最优性条件等。算法上，构造新的内点映射，定义中心路径并研究其代数与几何性质；基于核函数和代数等价变换研究NSCP的内点算法。应用上，利用矩阵分析、随机分析和凸松弛理论等，建立随机线性二次控制和H2/Hinf控制的精确半定规划松弛理论并设计内点算法求解。作为对称锥规划的理论和方法的推广，利用T-代数研究线性齐次锥规划的内点算法。本项目属最优化、基础数学、最优控制和计算机科学的交叉与融合，它的实施不仅能丰富锥规划理论，而且为最优控制提供新理论和新方法。

中文关键词： 对称锥规划；齐次锥规划；半定规划；内点算法；最优控制

英文摘要： Nonlinear symmetric cone programming，the nonlinear programming over symmetric cones, is the current frontier and international hot issues in optimization and has wide applications in optimal control, economic management, engineering technology, etc.. Interior-point algorithms have been successful applied to linear symmetric cone programming in both theory and practice. However, there is still a lack of work on interior-point algorithms for nonlinear symmetric cone programming. The purpose of the project is to study interior-point algorithms for nonlinear symmetric cone programming and applications in optimal control. In theory, we develop duality theory and optimality conditions for nonlinear symmetric cone programming by using Euclidean Jordan algebras and T-algebra, respectively. In algorithms, we construct the new interior-point mappings to define the central path and study its algebraic and geometric properties. Then, we design and analyze primal-dual interior-point algorithms for nonlinear symmetric cone programming based on eligible kernel functions and algebraically equivalent transformations, respectively. In applications, we develop the semidefinite programming relaxation theory for stochastic linear quadratic optimal control and H2/Hinf control by using matrix analysis, stochastic analysis and convex programming relaxation theory. Then, we solve the world optimal control problems by using kernel-based interior-point algorithms. As the generalizations of the theory and algorithms for symmetric cone programming, we also consider the kernel-based and full Nesterov-Todd step interior-point algorithms for homogeneous cone programming. This project is one of the challenging, fundamental and key problems in optimization, which relates to optimization, fundamental mathematics, optimal control and computer science. It has important scientific significance and great practical application value, which not only enriches the theory for cone programming, but also provides new theory and methods for optimal control.

英文关键词： Symmetric Cone Programming;Homogeneous Cone Programming;Semidefinite Programming;Interior-Point Algorithm;Optimal Control