MPVCC的约束规范及其应用
发布时间：2018-08-03 09:09
【摘要】：带有垂直互补约束的数学规划(MPVCC)问题是一种新形式的均衡优化模型,它涵盖了经典非线性规划模型和带有互补约束的数学规划(MPCC)问题并广泛地应用于经济均衡、工程、运输和网络设计等诸多行业范畴。由于非线性规划问题中绝大多数经典的约束规范在这一类问题中没有办法得到满足,因此,一般意义下的Karush-Kuhn-Tucker(KKT)条件不一定是MPVCC问题中的一阶必要性条件,这就使得MPVCC有多种形式的稳定点条件,如M-稳定点条件,S-稳定点(强稳定点)条件,W-稳定点(弱稳定点)条件等。众所周知,最优性条件是设计求解MPVCC算法的前提,因此寻找保证MPVCC稳定点条件的恰当的约束规范具有重要的理论意义和应用价值。本文主要关注MPVCC问题的约束规范及其应用,具体研究内容如下:首先,本文在总结MPVCC已有约束规范的基础上,给出了几个新的约束规范条件,例如MPVCC常秩约束规范、MPVCC常正约束规范、MPVCC-弱常正约束规范并研究了它们与MPVCC问题中已有约束规范之间的强弱关系。其次,把MPVCC的新的约束规范应用到一类MPVCC松弛方法的收敛性理论中。具体地,本文提出一种求解MPVCC问题的松弛方法,并在新的MPVCC约束规范条件下建立了算法的收敛性理论。最后,把一个基于博弈论的经济模型构造成MPVCC模型,然后利用松弛方法求解这个模型,数值算例验证了松弛方法的有效性。
[Abstract]:The (MPVCC) problem of mathematical programming with vertical complementary constraints is a new form of equilibrium optimization model. It covers the classical nonlinear programming model and the (MPCC) problem of mathematical programming with complementary constraints and is widely used in economic equilibrium and engineering. Transportation and network design and many other industries. Because most of the classical constraint specifications in nonlinear programming problems can not be satisfied in this kind of problems, the Karush-Kuhn-Tucker (KKT) condition in general sense is not necessarily the first order necessary condition in the MPVCC problem. This makes MPVCC have many kinds of stable point conditions, such as M- stable point condition and S- stable point condition (strong stable point) condition and W-stable point condition (weakly stable point) condition, and so on. It is well known that the optimality condition is the premise of designing the MPVCC algorithm. Therefore, it is of great theoretical significance and practical value to find the appropriate constraint specification to guarantee the MPVCC stability point condition. This paper mainly pays attention to the constraint specification of MPVCC problem and its application. The specific research contents are as follows: firstly, based on summarizing the existing constraint specification of MPVCC, several new constraint specification conditions are given. For example, the MPVCC constant rank constraint specification and the MPVCC-weakly constant positive constraint specification are used to study the strong and weak relationship between them and the existing constraints in the MPVCC problem. Secondly, the new constraint criterion of MPVCC is applied to the convergence theory of a class of MPVCC relaxation methods. In this paper, a relaxation method for solving MPVCC problem is presented, and the convergence theory of the algorithm is established under the condition of new MPVCC constraint specification. Finally, an economic model based on game theory is constructed into a MPVCC model, and then the relaxation method is used to solve the model. A numerical example is given to verify the effectiveness of the relaxation method.
【学位授予单位】：辽宁师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O221
本文编号：2161274
[Abstract]:The (MPVCC) problem of mathematical programming with vertical complementary constraints is a new form of equilibrium optimization model. It covers the classical nonlinear programming model and the (MPCC) problem of mathematical programming with complementary constraints and is widely used in economic equilibrium and engineering. Transportation and network design and many other industries. Because most of the classical constraint specifications in nonlinear programming problems can not be satisfied in this kind of problems, the Karush-Kuhn-Tucker (KKT) condition in general sense is not necessarily the first order necessary condition in the MPVCC problem. This makes MPVCC have many kinds of stable point conditions, such as M- stable point condition and S- stable point condition (strong stable point) condition and W-stable point condition (weakly stable point) condition, and so on. It is well known that the optimality condition is the premise of designing the MPVCC algorithm. Therefore, it is of great theoretical significance and practical value to find the appropriate constraint specification to guarantee the MPVCC stability point condition. This paper mainly pays attention to the constraint specification of MPVCC problem and its application. The specific research contents are as follows: firstly, based on summarizing the existing constraint specification of MPVCC, several new constraint specification conditions are given. For example, the MPVCC constant rank constraint specification and the MPVCC-weakly constant positive constraint specification are used to study the strong and weak relationship between them and the existing constraints in the MPVCC problem. Secondly, the new constraint criterion of MPVCC is applied to the convergence theory of a class of MPVCC relaxation methods. In this paper, a relaxation method for solving MPVCC problem is presented, and the convergence theory of the algorithm is established under the condition of new MPVCC constraint specification. Finally, an economic model based on game theory is constructed into a MPVCC model, and then the relaxation method is used to solve the model. A numerical example is given to verify the effectiveness of the relaxation method.
【学位授予单位】：辽宁师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O221
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