一类随机变分不等式的抽样平均近似方法.pdf

大连理工大学硕十学位论文摘要确定型变分不等式问题(Variational Inequality Problem)最初出现在数理方程中。1964 hia GS等学者首先建立了初期变分不等式(VIP)理论,随着变分不等式理论和运用的日益完善,变分不等式被广泛应用于工程,经济,交通运输, 运筹学与网络中。由于现实问题中往往存在许多不确定性因素,这样建立的变分不等式模型中往往卷入不确定参数,即称之为随机变分不等式(Stochastic Variational Inequality Problem)。随机变分不等式的研究刚刚引起学者关注,文献还不多见。抽样平均近似(Sample Average approximation)方法是利用蒙特卡洛模拟来求解随机优化的一种非常有力的方法,在这个技术中随机问题的期望目标函数通过平均样本来近似,然后利用确定性的优化方法对平均样本近似问题进行求解,从而得到原问题的一个近似解。本硕士论文在前人工作的基础上利用抽样平均近似方法对随机变分不等式问题进行研究。我利用Fukushima提出的正则化gap函数,给出了随机变分不等式的简单约束优化表达以及抽样平均近似问题,进而对抽样平均近似问题的最优值与最优解进行分析,为了验证所提出的抽样平均近似方法的有效性和可行性,我们对几个随机变分不等式实例进行了测试,数值结果表明我们的方法是可行的、有效的。关键词:随机变分不等式;抽样平均近似方法;正则化gap函数; 收敛性分析一类随机变分不等式的抽样平均近似方法 The Sample Average Approximation Method forthe Stochastic Variational Inequality Problem Abstract Variational Inequality Problems(In shortVIPs)firstly appeared inthePhysics equations. Lions J Land hia G Sfirstestablished the initialtheory ofVariational Inequality in thegreatimprovement in thetheory ofvafiational inequalities,VIPs have been effectively applied tomodding arange ofequilibrium problems inengineering,economics, transportation,operations research, many practicalproblems may involve uncertain factors,by modeling,models obtained are variational inequalities models where uncertainties callthem stochastic variational theliterature, many researersand practicers have begun to have increasing interesting instochastic variational inequalityproblems,but thereonly exist afew papers on stochastic variational inequalityproblems. The sample average approximation method is one ofeffective approachs for solving stochasticoptimizationproblems by using Monte Carlo thistechnique the expected objective function ofthestochasticproblem isapproximated by asample average ,thesample averageproblem issolvedby deterministicoptimization methods. The solution ofthesample average problem is areasonable approximation solution of the originalproblems. Onthebasis ofprevi