【摘要】：變分不等式問題(VIP)是運(yùn)籌學(xué)中的一個(gè)基本問題,同時(shí)在經(jīng)濟(jì)學(xué)、生態(tài)學(xué)、工程科學(xué)和金融學(xué)等很多領(lǐng)域具有廣泛應(yīng)用.因此,從上世紀(jì)60年代以來,變分不等式問題一直得到了眾多研究者的廣泛關(guān)注,特別是它的數(shù)值計(jì)算方法.比如人們熟知的算法有牛頓型算法、交替方向法、臨近點(diǎn)算法、內(nèi)點(diǎn)法、神經(jīng)網(wǎng)絡(luò)和投影法,其中,投影法以其簡單易操作的特點(diǎn),更是得到了眾多學(xué)者的青睞,對這類算法的研究層出不窮,何炳生提出的投影收縮算法就是其中的一類.投影收縮算法的特點(diǎn)是,每次的迭代計(jì)算量不大,一般是一些函數(shù)的簡單計(jì)算到可行集的投影.而本文的主要工作是對此進(jìn)行深一步研究,企圖構(gòu)造一種新的投影收縮算法來解決變分不等式問題.具體工作是改進(jìn)已有的投影收縮法,通過對何炳生的搜索方向進(jìn)行探索研究來考慮步長的選取,在原來的下降方向的前提下,得到了一個(gè)效率更高的步長,進(jìn)而得到一種新的投影收縮算法來解決變分不等式問題.具體內(nèi)容安排如下:第一章主要是緒論,首先對變分不等式問題的出現(xiàn)、發(fā)展以及其他一些背景知識進(jìn)行簡單的介紹,然后,給出了變分不等式問題和凸集上投影的一些基本概念和結(jié)論,接著,介紹了幾種常見的求解變分不等式問題的算法,最后,在章末給出了本文內(nèi)容上的安排工作.第二章首先給出了關(guān)于變分不等式問題的幾個(gè)不等式以及南京大學(xué)何炳生老師所提出的投影收縮算法的基本思想.其次,結(jié)合師兄王金龍的論文,對投影收縮算法的機(jī)理進(jìn)行了分析,提出了改進(jìn)線搜索策略的投影收縮算法,在原問題有解且向量值函數(shù)F單調(diào)的假設(shè)下,證明了新算法的全局收斂性.最后通過幾個(gè)算例,對改進(jìn)前后的兩種算法的數(shù)值實(shí)驗(yàn)結(jié)果進(jìn)行了對比,驗(yàn)證了改進(jìn)后的算法效率更高.第三章是對本文所做工作的總體分析和評價(jià),并對下一步研究工作進(jìn)行梳理和展望.
[Abstract]:Variational inequality problem (VIP) is a basic problem in operational research and has been widely used in many fields such as economics, ecology, engineering science and finance. Therefore, since the 1960s, variational inequality has been widely concerned by many researchers, especially its numerical method. For example, the well-known algorithms are Newton algorithm, alternating direction method, proximity point algorithm, interior point method, neural network and projection method. Among them, projection method is favored by many scholars because of its simple and easy to operate. There are many researches on this kind of algorithm, among which he Bingsheng's projection contraction algorithm is one of them. The characteristic of the projection contraction algorithm is that the computation of each iteration is small, and it is usually the projection from some simple functions to the feasible set. The main work of this paper is to study this problem one step further and try to construct a new projection contraction algorithm to solve the variational inequality problem. The specific work is to improve the existing projection contraction method, by exploring the search direction of he Bingsheng to consider the selection of step size. Under the premise of the original descent direction, a more efficient step is obtained. Then a new projection contraction algorithm is proposed to solve the variational inequality problem. The main contents are as follows: the first chapter is an introduction to the emergence, development and other background knowledge of variational inequalities. Some basic concepts and conclusions of variational inequality problems and projection on convex sets are given. Then, several common algorithms for solving variational inequality problems are introduced. At the end of the chapter, the arrangement work of this paper is given. In the second chapter, some inequalities about variational inequalities and the basic idea of projection contraction algorithm proposed by he Bingsheng of Nanjing University are given. Secondly, combining with Wang Jinlong's paper, the mechanism of projection contraction algorithm is analyzed, and a projection contraction algorithm with improved line search strategy is proposed. Under the assumption that the original problem is solved and the vector value function F is monotone, The global convergence of the new algorithm is proved. Finally, several examples are given to compare the numerical results of the two algorithms before and after the improvement, which proves that the improved algorithm is more efficient. The third chapter is the overall analysis and evaluation of the work done in this paper.
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O22

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本文編號：2404656