求解非凸随机二阶锥优化问题的无导数方法研究与应用

The stochastic second order conic optimization has wide application in engineering design, robot technology, classification technique, traffic management and financial engineering. At present, it is an important topic in the optimization and operations research. Based on the foundations of stochastic programming, stochastic process, second-order conic optimization and non-convex optimization, this project employs derivative free stochastic approximation (SA) method to solve non-smooth and non-convex stochastic second-order conic optimization.The main research contents are as follows. Firstly construct several functions which have important relationship with SA algorithm. Secondly, we propose derivative free SA algorithm and try to establish convergence analysis and complexity analysis. Finally, the algorithms in our project will be applied to grasping force optimization, investment portfolio and other fields. The anticipated results will play an active promoting role for the development of theory and numerical methods of stochastic conic optimization.

随机二阶锥优化问题在工程设计、机器人技术、分类技术、交通管理以及金融工程中都有广泛的应用，是数学规划领域的重要问题。本项目以随机优化、随机过程、二阶锥优化与非凸优化等理论为基础，研究求解非凸非光滑随机二阶锥优化问题的无导数随机近似（SA）算法及其应用。主要内容包括研究如何构造与二阶锥约束集合对应的几类重要函数，构造无导数SA算法并建立相应的算法收敛性和复杂性分析，并将所构造的算法应用于投资组合以及握力优化等实际应用领域。期望本项目对随机锥优化理论与数值方法的研究对该领域的发展起到积极的促进作用。

非凸随机二阶锥规划规划问题是一类新的、复杂的且具有广泛应用背景的优化问题，在工程、金融以及互联网领域都有着重要的应用。本项目以随机优化和二阶锥规划等理论为基础，结合凸分析、变分分析以及随机过程等知识，主要研究内容是如何应用无导数随机近似（SA）算法来求解带有二阶锥约束随机规划问题，包括构造与二阶锥约束集合相对应的几类关键函数；借助这些函数构造新的无导数SA算法并从理论的角度建立相应算法收敛性分析和复杂性分析；最后将所构造的算法应用于金融问题。本项目的研究结果拓宽了随机半定规划问题和SA算法的应用领域，具有重要的理论与实际意义。