【作者】 张杰；

【导师】 张立卫；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2011， 博士

【摘要】 由于客观地反映了实际中出现的随机因素,随机规划问题,尤其是随机均衡问题是目前最优化领域的研究热点。研究随机均衡问题的关键之一在于研究参数随机广义方程解映射的微分性质。本论文主要研究了参数随机广义方程的样本均值近似(SAA)解映射的微分性质及其应用,包括参数随机广义方程的SAA解映射的伴同导数的收敛性分析,带有约束的参数随机变分不等式的SAA解映射的伴同导数的收敛性分析及求解带有互补约束的随机规划的光滑化SAA方法。本论文所阐述的主要研究结果可概括如下：1.第三章研究了参数随机广义方程的SAA解映射的伴同导数的收敛性.由于伴同导数集合是无界的,本文引入了积分偏差和宇宙偏差来度量无界集合的偏差。在适当的条件下,证明了随着样本数的增加,SAA问题解映射的伴同导数到真问题的解映射的伴同导数的积分偏差和宇宙偏差以概率1收敛到0.其次,建立了SAA问题解映射的伴同导数到其真问题的解映射的伴同导数的指数收敛速率。最后把这些收敛性结果应用到SAA解映射类Lipschitz性质(或称Aubin性质)的相容性和带有互补约束的随机规划(SMPCC)问题的SAA估计式的稳定点的相容性分析中。2.第四章研究了带有随机等式和不等式约束的参数随机变分不等式问题的SAA解映射的伴同导数的收敛性.尽管本章研究的问题是参数随机广义方程的一种特殊形式,但SAA方法不同于前一章中的研究方法,因为参数随机变分不等式问题的约束部分的期望也需要用样本均值来近似.在适当的条件下,首先证明了随着样本数的增加,SAA问题解映射的伴同导数到真问题的解映射的伴同导数的以概率1的收敛性及指数收敛速率。然后应用收敛性结果来分析SAA解映射的类Lipschitz性质的相容性和SAA随机双层规划问题的稳定点的相容性。3.第五章提出了一类光滑化SAA方法来求解SMPCC问题。在适当的条件下,首先利用变分分析中上图收敛的概念讨论了光滑化SAA问题的最优解的以概率1收敛性。其次,证明了光滑化问题的Karush-Kuhn-Tucker (KKT)点序列的任何聚点都以概率1为SMPCC问题的一类稳定点。最后,在SMPCC的一类强二阶充分条件下,应用Robinson的非线性规划的稳定性理论,得到了光滑化SAA问题的KKT点序列的指数收敛速率。初步的数值结果表明了这种方法的有效性。更多还原

【Abstract】 As uncertain elements exist in many practical problems, the stochastic program prob-lems, especially the stochastic equilibrium problems have become an important concern in recent years. One of the key points in the study of stochastic equilibrium problems lies in the differential properties for the solution mapping governed by parametric stochastic generalized equations. This dissertation focuses on the differential properties of SAA solu-tion mappings to parametric stochastic generalized equations and applications, including convergence analysis of coderivatives of SAA solution mapping to a parametric stochastic generalized equation, convergence analysis of coderivatives of SAA solution map to a con-strained parametric stochastic variational inequalities and a smoothing SAA method for a stochastic mathematical program with complementarity constraints. The main results of this dissertation can be summarized as follows:1. In Chapter 3, convergence of coderivative of the SAA solution mapping to a para-metric stochastic generalized equation is studied. Because of the unboundedness of coderivative, the notions of cosmic deviation and the integrated deviation are introduced to estimate the deviation between unbounded sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated devi-ation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution mappings to the SAA parametric generalized equations is established. At last, the results are used to analyze the con-sistency of the Lipschitz-like property (or Aubin property) of the solution mapping of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.2. Chapter 4 focuses on the convergence analysis of coderivative of SAA solution mapping to a stochastic equality and inequality constrained parametric stochastic variational inequality. Although the model studied in this chapter is a special case of para-metric stochastic generalized equations, the SAA method is different from the one in Chapter 3 because the expectations in its constraints also need to be approx-imated by the sample averaging approach. Under suitable conditions, we at first show that the coderivative of the solution mapping to SAA problem exponentially converges to the coderivative of solution mapping to the parametric stochastic gen-eralized equation as the sample size tends to infinity with probability 1. And then, the results are used to analyze the consistency of the Lipschitz-like property of the solution mapping of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic bilevel program.3. In Chapter 5, a smoothing SAA method for a stochastic mathematical program with complementarity constraints (SMPCC) problem is discussed. Under suitable con-ditions, the almost sure convergence of the optimal solutions of the smoothed SAA problem is characterized by the notion of epi-convergence in variational analysis. It is also demonstrated that any accumulation point of Karash-Kuhn-Tucker points of the smoothed SAA problem is almost surely a kind of stationary point of SM-PCC as the sample size tends to infinity. At last, under a strong second-order sufficient condition for SMPCC, the exponential convergence rate of the sequence of Karash-Kuhn-Tucker points of the smoothed SAA problem is obtained through an application of Robinson’s stability theory. Some preliminary numerical results are reported to show the efficiency of the method proposed.更多还原