ERM方法求解随机线性二阶锥互补问题

发布时间：2018-01-02 02:34

本文关键词：ERM方法求解随机线性二阶锥互补问题　出处：《大连理工大学》2015年硕士论文　论文类型：学位论文

【摘要】：随机规划是对含有随机变量的优化问题进行建模的有效工具并已经拥有一个世纪的历史。二阶锥互补问题(SOCCP)是一类均衡优化问题。近年来,利用若当代数与谱分解,二阶锥互补问题的研究取得了重大进展。现阶段来说,二阶锥互补问题的理论研究与实际应用研究均呈现上升趋势,研究方向主要包括：解决二阶锥互补问题的各种光滑化方法,解的存在性与收敛性特征,以及其实际应用方向。因为问题通常具有各种不确定性,所以带有随机因素的二阶锥互补问题越来越多地受到人们的重视。本文引入了期望残差最小化(ERM)方法来求解随机线性二阶锥互补问题。本文主要研究了利用期望残差最小化方法求解随机线性二阶锥互补问题的解的存在性与收敛性,主要包括以下五个部分：第一部分,简要介绍背景材料。主要包括随机规划的产生与发展、随机互补问题的模型,二阶锥互补问题的模型、研究现状与实际应用方向。第二部分,给出一些预备知识。主要包括欧几里得若当代数的定义与主要性质、谱分解定理及收敛性证明中所需的引理。第三部分,介绍了四种互补问题的经典算法,并给出将随机线性互补问题转化为确定性的问题解决的三种转化模型。第四部分,利用ERM方法求解随机线性二阶锥互补问题。通过二阶锥互补函数FB函数进行问题的转化,将随机线性二阶锥互补问题转化为极小化问题,最后在进行合理假设的情况下证明离散型目标函数解的存在性与收敛性。最后,得出利用期望残差最小化方法解随机线性二阶锥互补问题,其离散型目标函数的解是存在且收敛的。
[Abstract]:Stochastic programming is an effective tool for modeling optimization problems with random variables and has a history of one century. The second order cone complementarity problem (SOCCP) is a class of equilibrium optimization problems. Great progress has been made in the study of the second-order cone complementarity problem using the contemporary number and spectral decomposition. At present, the theoretical research and practical application of the second-order cone complementarity problem are on the rise. The main research directions include: various smoothing methods for solving second-order cone complementarity problems, the existence and convergence characteristics of solutions, and their practical application directions, because the problems usually have various uncertainties. Therefore, more and more attention has been paid to the second order cone complementarity problem with random factors. In this paper, we introduce the expected residual minimization (ERM). In this paper, we study the existence and convergence of solutions to stochastic linear second-order cone complementarity problems by using the expected residual minimization method. It mainly includes the following five parts: the first part briefly introduces the background materials, mainly includes the generation and development of stochastic programming, the model of stochastic complementarity problem and the model of second-order cone complementarity problem. In the second part, some preliminary knowledge is given, including the definition and main properties of Euclidean number, the Lemma of spectral decomposition theorem and the proof of convergence. This paper introduces four classical algorithms for complementarity problems, and gives three transformation models for transforming stochastic linear complementarity problems into deterministic problems. Part 4th. The ERM method is used to solve the stochastic linear second-order cone complementarity problem, and the stochastic linear second-order cone complementarity problem is transformed into a minimization problem through the transformation of the second-order cone complementarity function FB function. Finally, the existence and convergence of the solution of discrete objective function are proved under reasonable assumptions. Finally, the expected residual minimization method is used to solve the stochastic linear second-order cone complementarity problem. The solution of discrete objective function exists and converges.
【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

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