基于高效蒙特卡罗策略的最优化方法及应用研究

项目名称： 基于高效蒙特卡罗策略的最优化方法及应用研究

项目编号： No.11501320

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 邵伟

作者单位： 曲阜师范大学

项目金额： 18万元

中文摘要： 蒙特卡罗方法是当今统计界重要的研究课题之一。由于蒙特卡罗方法对数据维数不敏感，使得用蒙特卡罗方法分析高维数据越来越引起人们的关注。本项目致力于高效蒙特卡罗方法的研究和设计，及其在高维数据的最优化问题中的应用。主要研究内容为：首先，在综合改进多测试Metropolis算法、不可逆马尔科夫链蒙特卡罗方法、自适应算法等多种加速策略的基础上，设计高效的马尔科夫链蒙特卡罗方法，使之适应于高维数据的同时并提高计算效率。其次，在上述高效蒙特卡罗方法的基础上，应用模拟退火算法寻找高维矩阵中较大平均值子矩阵，通过真实数据和模拟数据的例子比较模拟退火算法同已有算法的效率改进。最后，本项目还对高效蒙特卡罗方法在统计深度近似计算进行研究。研究并改进蒙特卡罗方法并高效的应用于最优化问题的分析和计算，不仅可以推进最优化方法在高维数据中的应用，同时还可以丰富高效蒙特卡罗方法的理论研究内容。

中文关键词： 蒙特卡罗策略；统计深度；随机模拟；高维数据；最优化方法

英文摘要： Monte Carlo methods is one of the important research topic in the statistical community today. Since Monte Carlo methods are not sensitive to the data dimension, more and more attention has been drawn to using the Monte Carlo methods to analyze high dimensional data. Our project is committed to research and design efficient Monte Carlo methods and their applications in the optimization problems in high dimensional data analysis. The main contents are as follows: Firstly, combining with various acceleration strategys, such as Multi-try Metropolis improved algorithm, non-reversible Markov Chain Monte Carlo, adaptive algorithm, we design efficient Markov Chain Monte Carlo methods to adapt the high-dimensional data, and improve the computational efficiency. Secondly, combining with these efficient Monte Carlo methods, we use simulated annealing algorithm to find the larger average submatrices in high dimensional matrix, then compare the efficiency improvement of the simulated annealing algorithm with that of some existing algorithms through real and simulated data examples. Thirdly, efficient Monte Carlo methods are used in the approximate computation of statistical depth. The research and improvment of Monte Carlo methods and their efficient applications in the analysis and computation of the optimization problems, not only promote the theory development of the applications of optimization methods in high dimensional data analysis, but also provide guidance to the theory of efficient Monte Carlo methods.

英文关键词： Monte Carlo strategies;Statistical depth;Stochastic simulation;High dimensional data;Optimization methods