本文選題：相關(guān)矩陣問(wèn)題 + 半光滑牛頓法�。� 參考：《湖南大學(xué)》2015年碩士論文

【摘要】：最佳相關(guān)矩陣問(wèn)題是指在Frobenius范數(shù)下尋找與給定的對(duì)稱矩陣最接近的相關(guān)矩陣.最佳相關(guān)矩陣問(wèn)題一般有不帶權(quán)、帶W權(quán)、帶H權(quán)、帶Q權(quán)等類型.本文主要針對(duì)前三種類型的理論和數(shù)值解法展開(kāi)進(jìn)一步的研究.本文首先介紹了最佳相關(guān)矩陣問(wèn)題的研究現(xiàn)狀和進(jìn)展,在此基礎(chǔ)上提出了本文的工作構(gòu)想.同時(shí),為方便后面的研究,本文給出了最優(yōu)化的一些基礎(chǔ)知識(shí)以及相關(guān)的優(yōu)化算法.在第二章,我們首先研究了W權(quán)問(wèn)題的特殊形式—不帶權(quán)問(wèn)題的理論與數(shù)值解法.對(duì)于不帶權(quán)問(wèn)題,我們分析了它與它的對(duì)偶問(wèn)題之間的關(guān)系—原問(wèn)題的解可用其對(duì)偶問(wèn)題的解表示.由于對(duì)偶問(wèn)題等價(jià)于一個(gè)半光滑方程組,在求解半光滑方程組的牛頓法的基礎(chǔ)上,利用正則化策略修改牛頓方程,我們提出了求解半光滑方程組的正則化牛頓法.它的優(yōu)點(diǎn)是計(jì)算出的搜索方向一定是目標(biāo)函數(shù)的下降方向,有效克服了半光滑牛頓法的固有缺陷.為了提高正則化牛頓法的效率,我們研究了求解牛頓方程的共軛梯度法(即內(nèi)層迭代)的預(yù)處理策略、內(nèi)層迭代控制變量取值的優(yōu)化處理,然后我們提出了一個(gè)改善的正則化牛頓法,并分析了它的全局收斂性和二次收斂速度.最后,我們研究了改善的正則化牛頓法的計(jì)算解的相關(guān)性處理,使得最終計(jì)算出的解更接近相關(guān)矩陣,并分析了它與最優(yōu)解的誤差估計(jì).由于一般的W權(quán)問(wèn)題可以通過(guò)簡(jiǎn)單的變換轉(zhuǎn)化為不帶權(quán)問(wèn)題,因此上面關(guān)于不帶權(quán)問(wèn)題的研究工作可推廣到一般的W權(quán)問(wèn)題的求解.在第三章,我們分析了帶H權(quán)問(wèn)題的理論和數(shù)值解法.首先我們給出了這類問(wèn)題的約束非退化性質(zhì)和強(qiáng)二階充分條件,以及廣義Jacobi的計(jì)算公式,然后在此基礎(chǔ)上我們對(duì)求解H權(quán)問(wèn)題的牛頓—共軛梯度法(Newton-CG法)進(jìn)行了改善.在內(nèi)層迭代中,利用目標(biāo)函數(shù)在迭代點(diǎn)處的一階、二階信息對(duì)控制變量的取值進(jìn)行重新定義,均衡內(nèi)外層迭代的計(jì)算量,有效提高算法的效率.通過(guò)簡(jiǎn)單的分析得出改善后的Newton-CG法仍然具有二次收斂性.文章最后對(duì)本文提出的所有算法進(jìn)行了數(shù)值測(cè)試.數(shù)值實(shí)驗(yàn)結(jié)果表明,本文所提出的算法具有很好的數(shù)值效果.
[Abstract]:The optimal correlation matrix problem is to find the most close correlation matrix to the given symmetric matrix under Frobenius norm. There are three kinds of optimal correlation matrix problems: no weight, W weight, H weight, Q weight and so on. This paper focuses on the first three types of theoretical and numerical solutions to further research. This paper first introduces the research status and progress of the optimal correlation matrix problem, and then puts forward the working conception of this paper. At the same time, in order to facilitate the following research, this paper gives some basic knowledge of optimization and related optimization algorithms. In the second chapter, we first study the special form of W weight problem, the theory and numerical solution of unweighted problem. For the unweighted problem, we analyze the relationship between it and its dual problem, the solution of the original problem can be represented by the solution of its dual problem. Since the duality problem is equivalent to a semi-smooth system of equations, on the basis of Newton's method for solving semi-smooth equations, we propose a regularized Newton method for solving semi-smooth equations by using regularization strategy to modify Newton's equations. Its advantage is that the calculated search direction must be the descent direction of the objective function, which effectively overcomes the inherent defects of the semi-smooth Newton method. In order to improve the efficiency of regularization Newton method, we study the preprocessing strategy of conjugate gradient method (I. E. inner iteration) for solving Newton equation, and optimize the value of inner iteration control variable. Then we propose an improved regularization Newton method and analyze its global convergence and quadratic convergence rate. Finally, we study the correlation treatment of the computational solution of the improved regularized Newton method, which makes the final solution closer to the correlation matrix, and analyzes the error estimates between the solution and the optimal solution. Since a general W weight problem can be transformed into a non weighted problem by a simple transformation, the above research work on the non weighted problem can be extended to the solution of the general W weight problem. In the third chapter, we analyze the theory and numerical solution of H-weighted problem. First of all, we give the constrained nondegenerate property, strong second order sufficient condition and generalized Jacobi formula for this kind of problems. Then we improve the Newton-CG method for solving H-weight problems. In the inner layer iteration, the first order and second order information of the objective function at the iteration point is used to redefine the value of the control variable, to balance the computation of the inner and outer layer iteration, and to improve the efficiency of the algorithm effectively. A simple analysis shows that the improved Newton-CG method still has quadratic convergence. Finally, all the algorithms proposed in this paper are numerically tested. The numerical results show that the proposed algorithm has a good numerical effect.
【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O224

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本文編號(hào)：2099650