求解随机二阶锥互补问题的期望值与样本均值近似方法

发布时间：2018-05-07 17:09

本文选题：随机 + 二阶锥互补问题　；参考：《辽宁大学》2017年硕士论文

【摘要】：二阶锥互补问题(SOCCP)是指在二阶锥约束条件下两组决策变量之间满足一种“互补”关系,是一类均衡优化问题.借助于欧几里得若当代数技术,近年来SOCCP得到了快速的发展.二阶锥互补问题(SOCCP)在经济、工程等领域都有着广泛的应用.然而,现实生活中会存在一些不确定因素,忽视这些因素将会使决策失误.为此,本文考虑随机二阶锥互补问题(SSOCCP).由于随机变量的存在,随机二阶锥互补问题一般情况下无解.然而为了满足含有随机因素的实际问题对解的迫切要求,这需要我们构造一个合理的确定性模型,再对该确定模型进行求解,并将该确定模型的解视为随机二阶锥互补问题的解.因此,为了得到随机二阶锥互补问题的合理的解,本文利用二阶锥互补函数给出求解随机二阶锥互补问题的确定期望值(EV)模型.本文分别考虑应用二阶锥互补函数ΦT及ΦNR给出EV模型,并首先给出了该EV模型水平集有界的条件.当二阶锥互补函数为ΦT时,本文首先讨论了 EV模型目标函数的SC1性.由于该EV模型中含有数学期望,而期望不容易求得.为求解此模型,本文应用样本均值近似(SAA)方法给出此模型的近似问题.在理论上,本文进一步考虑了EV模型近似问题全局最优解序列以及稳定点序列的收敛性结果.当二阶锥互补函数为ΦNR时,由于此时对应的EV模型的目标函数是非光滑的,本文先利用光滑化方法给出相应目标函数的光滑化函数,并进一步应用SAA方法给出近似问题.与ΦT对应的EV模型类似,在理论上本文依然给出了当二阶锥互补函数为ΦNR时,全局最优解序列以及稳定点序列的收敛性分析.最后,本文给出数值算例,并分别应用所提方法求解.
[Abstract]:The second order cone complementarity problem (SOCCP) is a kind of "complementary" relation between two sets of decision variables under the condition of second-order conical constraint, and it is a kind of equilibrium optimization problem. With the aid of Euclidean modern numerical technology, SOCCP has been developed rapidly in recent years. Second order Cone complementarity problem (SOCCP) is widely used in economy, engineering and other fields. However, there are some uncertain factors in real life, and neglecting these factors will make decision-making mistakes. In this paper, we consider the stochastic second order cone complementarity problem. Because of the existence of random variables, there is no solution for the random second order cone complementarity problem. However, in order to meet the urgent need for the solution of practical problems with random factors, we need to construct a reasonable deterministic model and then solve the deterministic model. The solution of the model is regarded as the solution of the random second order cone complementarity problem. Therefore, in order to obtain a reasonable solution of the stochastic second-order cone complementarity problem, a definite expected value (EV) model for solving the stochastic second-order cone complementarity problem is given by using the second-order cone complementarity function. In this paper, the second order cone complementary function 桅 T and 桅 NR are used to give the EV model, and the bounded condition of the level set of the EV model is given. When the second order cone complementary function is 桅 T, we first discuss the SC1 property of the objective function of EV model. Because the EV model contains mathematical expectation, the expectation is not easy to get. In order to solve this model, the approximate problem of the model is given by using the sample mean approximation (SAA) method. In theory, the convergence results of the sequence of global optimal solutions and the sequence of stable points for the EV model approximation problem are further considered in this paper. When the second order cone complementary function is 桅 NR, the objective function of the corresponding EV model is non-smooth at this time. In this paper, the smoothing function of the corresponding objective function is given by using the smoothing method, and the approximate problem is given by using the SAA method. Similar to the EV model corresponding to 桅 T, the convergence analysis of the global optimal solution sequence and the stable point sequence is given when the second order cone complementary function is 桅 NR. Finally, numerical examples are given and solved by the proposed method.
【学位授予单位】：辽宁大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O221

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