一类分布鲁棒二次规划问题

发布时间：2018-06-18 02:22

本文选题：分布鲁棒优化 + 最小方差　；参考：《大连理工大学》2015年硕士论文

【摘要】：分布鲁棒优化问题(Distributionally Robust Optimization Problems)是建立在考虑最坏情况下进行优化的所谓鲁棒优化问题(Robust Optimization Problems)的基础上,统筹考虑投资组合中的不确定性.作为Markowitz期望-方差模型分布鲁棒优化形式的扩展,本文讨论了一类分布鲁棒二次规划问题,问题的目标是在最坏的风险不大于一定值的约束下,最大化最坏情况下的收益.该鲁棒优化模型的分布集合是由随机变量的期望和方差的上界来确定的.首先利用对偶定理证明了这类问题可以转化为易解的凸二次SDP规划问题,因而可以用Matlab的通用程序求解.我们给出了一些实际算例,数值结果验证了该方法解决分布鲁棒二次规划问题的可行性.本文的内容概括如下：1.第一章介绍了Markowitz期望-方差模型的产生与发展及分布鲁棒优化问题的背景,研究现状,然后提出本文所研究的一类分布鲁棒二次规划问题.2.第二章介绍了一些矩阵,概率论,对偶理论等基础知识,更有助于本文模型的理解.3.第三章证明了本文讨论的一类分布鲁棒二次规划问题等价于一个易解的凸二次SDP规划问题,这一结论通过两次运用Lagrange对偶方法得出.4.第四章首先用MATLAB工具箱YALMIP进行数值试验,我们得到了很好的数值结果.然后我们对模型进行深入的分析,发现分布鲁棒优化模型能够很好的规避风险.最后我们研究了双分布鲁棒优化模型以及目标函数比较简单的单分布鲁棒优化模型,并将两个模型进行了对比分析,最后,我们引入无风险资产,并写出了新的模型.
[Abstract]:Distributed robust Optimization problem (DROP) is a robust optimization problem based on robust optimization in the worst-case scenario, and the uncertainty in the portfolio is considered as a whole. As an extension of the distributed robust optimization form of Markowitz's expected variance model, this paper discusses a class of distributed robust quadratic programming problems. The goal of the problem is to maximize the return in the worst case under the constraint that the worst risk is not greater than a certain value. The distribution set of the robust optimization model is determined by the expectation of random variables and the upper bound of variance. Firstly, it is proved that this kind of problem can be transformed into a convex quadratic SDP programming problem by using duality theorem, so it can be solved by the general program of Matlab. Some practical examples are given and the numerical results show that the proposed method is feasible for solving the distributed robust quadratic programming problem. The content of this article is summarized as follows: 1. In the first chapter, we introduce the background of Markowitz's expected variance model and the background of distributed robust optimization problem. Then we propose a class of distributed robust quadratic programming problem. The second chapter introduces some basic knowledge, such as matrix, probability theory, duality theory and so on, which is helpful to the understanding of this model. In chapter 3, we prove that a class of distributed robust quadratic programming problems discussed in this paper is equivalent to an easily solvable convex quadratic SDP programming problem. This conclusion is obtained by using Lagrange duality method twice. In the fourth chapter, we use the MATLAB toolbox YALMIP to carry on the numerical experiment, we get the very good numerical result. Then we analyze the model and find that the distributed robust optimization model can avoid risk. Finally, we study the dual distribution robust optimization model and the single distributed robust optimization model with simple objective function, and compare the two models. Finally, we introduce the riskless assets and write out a new model.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O221

【共引文献】