求解随机二阶锥互补问题的条件风险价值模型及其收敛性分析

发布时间：2019-01-05 09:32

【摘要】：二阶锥互补问题(SOCCP)作为一类具有普遍意义的均衡优化问题,近年来备受关注.学者们将欧几里得若当代数与谱分解作为工具,使得二阶锥互补问题的研究取得重大进展.目前,有关确定性二阶锥互补问题的研究在理论和实际应用方面都有良好的发展趋势,理论方面的主要研究方向有:SOCCP的各类光滑化问题的求解、有关解的存在性与收敛性分析以及一些有效算法的开发研究等.实际应用方面,许多问题均可转化为二阶锥互补问题,例如三维摩擦接触问题、鲁棒纳什均衡问题等.然而,实际问题往往包含诸如价格、供应、需求等不确定因素,忽略这些随机因素将导致严重的后果.由于随即变量的引入,使得随机二阶锥互补问题比二阶锥互补问题更复杂,进而在实际方面也有更加广泛的应用.因此,随机二阶锥互补问题的相关研究是十分必要且意义重大.基于上述原因,本文提出了求解随机二阶锥互补问题的条件风险价值(CVaR)模型.本文将二阶锥互补函数作为损失函数,结合光滑化方法和蒙特卡罗样本均值近似方法给出随机二阶锥互补问题(SSOCCP)的CVaR光滑化模型以及CVaR光滑化样本均值近似模型,进一步对对应光滑化问题及光滑近似问题进行收敛性分析,并且给出相关的数值算例并应用所提出的方法进行求解.最后,对本文所研究的主要内容进行详细的总结,并对SSOCCP的条件风险价值模型提出进一步的假设和展望.
[Abstract]:The second order cone complementarity problem (SOCCP), as a universal equilibrium optimization problem, has attracted much attention in recent years. By using Euclidean number and spectral decomposition as tools, great progress has been made in the study of second-order cone complementarity problem. At present, the research on deterministic second-order cone complementarity problem has a good development trend in theory and practical application. The main research directions of theory are: solving various smoothing problems of SOCCP. The existence and convergence of solutions and the development of some effective algorithms are discussed. In practical applications, many problems can be transformed into second-order cone complementarity problems, such as three-dimensional frictional contact problems, robust Nash equilibrium problems and so on. However, practical problems often include uncertain factors such as price, supply, demand and so on. Ignoring these random factors will lead to serious consequences. Because of the introduction of random variables, the random second-order cone complementarity problem is more complex than the second-order cone complementarity problem, and it is also widely used in practice. Therefore, it is necessary and significant to study the random second order cone complementarity problem. Based on the above reasons, this paper presents a conditional risk value (CVaR) model for solving stochastic second-order cone complementarity problems. In this paper, the second order cone complementary function is taken as the loss function, and the CVaR smoothing model and the CVaR smoothing sample mean approximation model for the stochastic second order cone complementarity problem (SSOCCP) are given by combining the smoothing method and the Monte Carlo sample mean approximation method. Furthermore, the convergence of the corresponding smoothing problem and the smooth approximation problem is analyzed, and the numerical examples are given and solved by the proposed method. Finally, the main contents of this paper are summarized in detail, and further hypotheses and prospects for the conditional risk value model of SSOCCP are put forward.
【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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