【摘要】：帶有垂直互補約束的數(shù)學規(guī)劃(MPVCC)問題是一種新形式的均衡優(yōu)化模型,它涵蓋了經(jīng)典非線性規(guī)劃模型和帶有互補約束的數(shù)學規(guī)劃(MPCC)問題并廣泛地應用于經(jīng)濟均衡、工程、運輸和網(wǎng)絡設計等諸多行業(yè)范疇。由于非線性規(guī)劃問題中絕大多數(shù)經(jīng)典的約束規(guī)范在這一類問題中沒有辦法得到滿足,因此,一般意義下的Karush-Kuhn-Tucker(KKT)條件不一定是MPVCC問題中的一階必要性條件,這就使得MPVCC有多種形式的穩(wěn)定點條件,如M-穩(wěn)定點條件,S-穩(wěn)定點(強穩(wěn)定點)條件,W-穩(wěn)定點(弱穩(wěn)定點)條件等。眾所周知,最優(yōu)性條件是設計求解MPVCC算法的前提,因此尋找保證MPVCC穩(wěn)定點條件的恰當?shù)募s束規(guī)范具有重要的理論意義和應用價值。本文主要關注MPVCC問題的約束規(guī)范及其應用,具體研究內(nèi)容如下:首先,本文在總結MPVCC已有約束規(guī)范的基礎上,給出了幾個新的約束規(guī)范條件,例如MPVCC常秩約束規(guī)范、MPVCC常正約束規(guī)范、MPVCC-弱常正約束規(guī)范并研究了它們與MPVCC問題中已有約束規(guī)范之間的強弱關系。其次,把MPVCC的新的約束規(guī)范應用到一類MPVCC松弛方法的收斂性理論中。具體地,本文提出一種求解MPVCC問題的松弛方法,并在新的MPVCC約束規(guī)范條件下建立了算法的收斂性理論。最后,把一個基于博弈論的經(jīng)濟模型構造成MPVCC模型,然后利用松弛方法求解這個模型,數(shù)值算例驗證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.
【學位授予單位】：遼寧師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O221

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