多目标优化及随机变分不等式问题的若干研究

发布时间：2018-02-28 12:22

本文关键词： 多目标优化问题强Karush-Kuhn-Tucker条件凸化集随机变分不等式拟蒙特卡洛方法　出处：《重庆大学》2016年博士论文　论文类型：学位论文

【摘要】：本文主要研究了以Clarke次微分和凸化集刻画的非光滑多目标优化问题的强Karush-Kuhn-Tucker条件及随机变分不等式问题的加权期望残差法。全文共分为七章,具体如下:第一章,回顾了多目标优化问题强Karush-Kuhn-Tucker条件的研究现状,阐述了随机优化问题的研究概况,并介绍了本文的选题动机和主要内容。第二章,介绍了本文后面经常用到的一些定义、符号以及性质,主要包括非线性优化问题中常见的约束品性以及切锥、法锥、期望、密度函数等概念。第三章,考虑带有等式、不等式和集合约束的多目标优化问题的强Karush-Kuhn-Tucker条件,其中目标函数和约束函数均为Lipschitz连续。首先,引入与目标函数和约束系统均相关的平静性条件,并且证明了其等价于两种精确罚问题。然后,基于上述结论,建立了多目标优化问题在局部弱有效解处以Clarke次微分刻画的强Karush-Kuhn-Tucker条件。最后,在光滑的情况下,借助多目标优化问题平静性条件和局部误差界性质之间的关系,研究了广义的Mangasarian-Fromovitz约束品性和多目标优化问题的平静性条件之间的关系。第四章,针对带有不等式约束和集合约束的非光滑多目标优化问题,引入两种约束品性,即(CQ1)和(CQ2)。首先,在(CQ1)和(CQ2)下,分别建立了非光滑多目标优化问题在局部有效解处以凸化集刻画的强Karush-Kuhn-Tucker条件。其次,举例说明了强Karush-Kuhn-Tucker条件中出现的凸包和闭包不能去掉,并且条件中目标函数的上半正则凸化集不可弱化为上凸化集。最后,比较了(CQ1)、(CQ2)和最近文献中出现的约束品性之间的关系。第五章,基于正则间隙函数的绝对值残差和最小二乘残差期望的凸组合,将带有非线性扰动的随机仿射变分不等式问题转化为一个加权期望残差极小化问题,并在一定的条件下得到了加权期望残差极小化问题的一些性质。此外,通过拟蒙特卡洛方法,给出加权期望残差极小化问题的离散近似问题,并对离散近似问题的最优解和稳定点进行了收敛性分析。第六章,针对随机非线性变分不等式问题,借助正则间隙函数的绝对值残差和最小二乘残差期望的凸组合方法,把随机非线性变分不等式问题转化为一个加权期望残差极小化问题。此外,针对样本空间为紧集的情况,通过拟蒙特卡洛方法得到了加权期望残差极小化问题的离散近似问题。针对样本空间为非紧集的情况,利用紧近似方法得到了加权期望残差极小化问题的紧近似问题,并分别对离散近似问题和紧近似问题的最优解进行了收敛性分析。第七章,对本文的主要内容做了简单的总结,并提出了一些值得思考和今后准备研究的问题。
[Abstract]:In this paper, we mainly study the strong Karush-Kuhn-Tucker conditions for nonsmooth multiobjective optimization problems characterized by Clarke subdifferential and convex sets and the weighted expected residuals method for stochastic variational inequality problems. This paper reviews the research status of strong Karush-Kuhn-Tucker conditions for multi-objective optimization problems, expounds the general situation of stochastic optimization problems, and introduces the motivation and main contents of this paper. Chapter two introduces some definitions that are often used in this paper. Symbols and properties, mainly including the common constraints in nonlinear optimization problems, as well as the tangent cone, normal cone, expectation, density function and other concepts. The strong Karush-Kuhn-Tucker conditions for multiobjective optimization problems with inequality and set constraints, in which the objective function and the constraint function are Lipschitz continuous, are introduced. Firstly, the calm conditions related to both the objective function and the constrained system are introduced. And it is proved that it is equivalent to two exact penalty problems. Then, based on the above conclusions, the strong Karush-Kuhn-Tucker conditions for the local weak efficient solutions of multiobjective optimization problems to be characterized by Clarke subdifferential are established. Finally, in the case of smoothness, With the help of the relationship between the calm conditions of multiobjective optimization problems and the properties of local error bounds, the relationship between generalized Mangasarian-Fromovitz constraint properties and calm conditions for multiobjective optimization problems is studied. Chapter 4th, For non-smooth multi-objective optimization problems with inequality constraints and set constraints, two constraints, namely, CQ1) and CQ2, are introduced. Firstly, under CQ1 and CQ2), In this paper, we establish the strong Karush-Kuhn-Tucker conditions for the locally efficient solutions of nonsmooth multiobjective optimization problems to be characterized by convexity sets. Secondly, we illustrate that convex hulls and closures in strong Karush-Kuhn-Tucker conditions can not be removed. Moreover, the upper semi-regular convex set of objective function can not be reduced to the upper convex set. Finally, the relationship between CQ1 / CQ2) and the constraint character in recent literature is compared. Chapter 5th, Based on the convex combination of absolute residual and least square residual expectation of regular gap function, the stochastic affine variational inequality problem with nonlinear perturbation is transformed into a weighted expected residual minimization problem. Some properties of the weighted expected residual minimization problem are obtained under certain conditions. In addition, the discrete approximation problem of the weighted expected residual minimization problem is given by using the quasi Monte Carlo method. In chapter 6th, for stochastic nonlinear variational inequality problems, the convex combination method of absolute residual and least square residual expectation of regular gap function is used to analyze the convergence of the optimal solution and the stable point of the discrete approximation problem. The stochastic nonlinear variational inequality problem is transformed into a weighted expected residual minimization problem. The discrete approximation problem of the weighted expected residual minimization problem is obtained by using the quasi-Monte Carlo method, and the compact approximation problem for the weighted expected residual minimization problem is obtained by using the compact approximation method, where the sample space is a non-compact set. The convergence of the optimal solutions of discrete approximation problems and compact approximation problems is analyzed respectively. Chapter 7th makes a brief summary of the main contents of this paper and puts forward some problems worth considering and studying in the future.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O178;O221.6

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