本文選題：非線性互補問題　切入點：ODE　出處：《河北大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：非線性互補問題是指這樣的問題:被其包含的兩組決策變量之間滿足一種互補關(guān)系,根據(jù)問題中變量所滿足的條件的不同,以及互補關(guān)系的不同形式,互補問題存在很多種類型,本文主要討論一般的非線性互補問題.這類問題有著廣泛的應(yīng)用背景,諸如純交換的競爭經(jīng)濟的均衡問題、具有生產(chǎn)和投資的經(jīng)濟均衡問題、工程中的電路聯(lián)結(jié)問題、流體力學(xué)問題、最優(yōu)控制問題等.因此對非線性互補問題求解具有重要意義.本文將非線性互補問題轉(zhuǎn)化為約束優(yōu)化問題,而對于約束優(yōu)化問題的求解,Fletcher和Leyffer曾提出了一類仿照多目標(biāo)優(yōu)化的濾子方法,在這類傳統(tǒng)的方法中搜索方向都是通過求解序列二次規(guī)劃子問題得到的,這使得算法運算量比較大.另一方面,傳統(tǒng)濾子方法中的點對只包含了目標(biāo)函數(shù)的信息,并沒有充分利用其他信息.考慮到上述二次子問題運算量大以及濾子點列信息利用不充分的問題,借助于常微分方程(ODE)方法,利用求解一個非線性方程來計算搜索方向,在改造的濾子集合基礎(chǔ)上,提出了求解非線性互補問題的兩種算法,一種是將NCP函數(shù)信息引入到濾子集合中,提出帶有一般NCP函數(shù)的ODE型濾子方法.另一種是將NCP函數(shù)光滑化,給出光滑化算子的更新公式,從而得到光滑化的ODE型濾子方法.本文中所提出的方法,與傳統(tǒng)的濾子方法相比,更為靈活且便于實現(xiàn),同時算法的計算量相對減少.在合理的假設(shè)前提下,我們證明了算法的全局收斂性.
[Abstract]:A nonlinear complementarity problem is a problem in which the two sets of decision variables contained therein satisfy a complementary relationship, depending on the conditions of the variables in the problem, as well as the different forms of the complementary relationship. There are many types of complementarity problems. This paper mainly discusses general nonlinear complementarity problems, which have a wide application background, such as the equilibrium problem of pure exchange competitive economy, the economic equilibrium problem with production and investment. The problems of circuit connection, hydrodynamics, optimal control and so on in engineering are very important for solving nonlinear complementarity problems. In this paper, nonlinear complementarity problems are transformed into constrained optimization problems. For constrained optimization problems, Fletcher and Leyffer have proposed a kind of filter method that mimics multi-objective optimization. In this kind of traditional method, the search direction is obtained by solving the sequential quadratic programming subproblem. On the other hand, the point pairs in the traditional filter method only contain the information of the objective function. Considering the problem that the quadratic subproblem has a large amount of computation and the information of filter points is not sufficiently utilized, the search direction is calculated by solving a nonlinear equation with the aid of ordinary differential equation (ODE) method. On the basis of the modified filter set, two algorithms for solving nonlinear complementarity problems are proposed. One is to introduce the information of NCP function into the filter set, and the other is to smooth the NCP function by introducing the information of ODE filter with general NCP function. The updating formula of smoothing operator is given, and the smoothing ODE filter method is obtained. Compared with the traditional filter method, the method presented in this paper is more flexible and convenient to realize. At the same time, the computational complexity of the algorithm is relatively reduced. On the premise of reasonable assumption, we prove the global convergence of the algorithm.
【學(xué)位授予單位】：河北大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O221.2

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本文編號：1643818