非线性方程组的锥模型方法研究

发布时间：2018-03-30 06:59

本文选题：非线性方程组　切入点：锥模型　出处：《内蒙古大学》2017年博士论文

【摘要】：随着科学技术的发展和计算机的广泛应用,非线性方程组问题越来越受到人们的关注,非线性方程组的求解问题也成为活跃的研究课题.它在人工智能、机器学习、金融计算、防灾研究、能源探测以及气象预报等各个邻域有着广泛的运用.本文主要对求解非线性方程组的锥模型方法进行研究.其中主要包括三类内容,第一是求解光滑非线性方程组的一类改进的锥模型牛顿法,第二是求解目标函数具有特殊结构的无约束优化问题的结构型拟牛顿法,第三是求解一类非光滑方程组的光滑型方法.所取得的主要结果有:1.分析了两点有理逼近模型算法和锥模型算法的关系,阐明了两点有理逼近模型算法是锥模型算法的特殊情形,由此对两点有理逼近算法的改进与完善提供了理论框架.2.提出了两点有理逼近模型的若干改进方法.首先,提出了更合理地筛选有理逼近解的方法并证明了该逼近的单调性.其次,对于原函数在当前点与前次迭代点连线方向上的方向导数符号相反的情况,分别提出了迭代求有理逼近和构造在当前点与估算点连线方向上相应的方向导数符号相同的近似有理逼近的方法.此外,提出了一个非单调的有理逼近函数.最后,通过数值计算验证了本文提出的改进方法是有效和可行的.3.提出了近似逼近向量值函数的一类特殊锥模型,基于此模型给出了求解非线性方程组的一种改进的锥模型牛顿算法.该算法的主要特点是每一步迭代都利用一个秩一矩阵修正Jacobi矩阵.在一般条件下证明了算法具有局部二阶收敛性.数值实验和对比表明了算法的有效性.4.提出了求解目标函数具有特殊结构的无约束优化问题的结构型锥拟牛顿算法.首先利用锥模型及其最近两次迭代点上的插值条件推导出了锥拟牛顿方程.标准拟牛顿方程中仅仅使用目标函数的梯度信息,而锥拟牛顿方程不仅利用目标函数的梯度信息还要用到目标函数的函数值信息.其次,基于锥拟牛顿方程提出了一类结构型锥拟牛顿算法.并证明了算法的局部超线性收敛性.该算法适合求解目标函数的Hesse矩阵有特殊结构和部分可利用信息的无约束优化问题,非线性最小二乘问题是该类问题的典型例子.5.提出了求解绝对值方程的一类光滑型算法,并比较了四个光滑化函数的数值表现.绝对值方程问题是一类不可微的NP-hard问题.基于新给出的光滑化函数,本文将绝对值方程转化成等价的光滑方程组,并应用相应的光滑型算法求解此方程组.我们的主要贡献在于数值实验和比较分析,通过数值比较不仅选出了四个光滑化函数中数值表现最好的函数,还给出了四个函数在迭代次数和计算时间方面的数值表现的排序.
[Abstract]:With the development of science and technology and the wide application of computer, people pay more and more attention to the problem of nonlinear equations, and the problem of solving nonlinear equations has become an active research topic.It is widely used in artificial intelligence, machine learning, financial computing, disaster prevention, energy detection and weather forecast.In this paper, the cone model method for solving nonlinear equations is studied.It mainly includes three kinds of contents: the first is an improved cone model Newton method for solving smooth nonlinear equations, the second is a structured quasi-Newton method for solving unconstrained optimization problems with special structure of objective function.The third is a smooth method for solving a class of nonsmooth equations.The main results achieved were: 1: 1.The relationship between two-point rational approximation model algorithm and cone model algorithm is analyzed. It is clarified that two-point rational approximation model algorithm is a special case of cone model algorithm, which provides a theoretical framework for the improvement and perfection of two-point rational approximation algorithm.Some improved methods of two-point rational approximation model are presented.Firstly, a more reasonable method for screening the solution of rational approximation is proposed and the monotonicity of the approximation is proved.Secondly, for the case where the original function has the opposite sign of directional derivative in the direction of the line line between the current point and the previous iteration point,In this paper, an iterative method for finding rational approximation and constructing approximate rational approximation with the same sign of directional derivative in the line direction of the current point and the estimated point are presented respectively.In addition, a nonmonotone rational approximation function is proposed.Finally, the numerical results show that the proposed improved method is effective and feasible.In this paper, a special cone model approximating vector-valued functions is proposed. Based on this model, an improved cone model Newton algorithm for solving nonlinear equations is presented.The main feature of the algorithm is that each iteration uses a rank one matrix to modify the Jacobi matrix.The local second order convergence of the algorithm is proved under general conditions.Numerical experiments and comparisons show that the algorithm is effective. 4.Firstly, the cone quasi Newton equation is derived by using the interpolation condition on the cone model and its two most recent iterations.The standard quasi Newton equation only uses the gradient information of the objective function, while the cone quasi Newton equation not only uses the gradient information of the objective function, but also uses the function value information of the objective function.Secondly, a class of structural cone quasi-Newton algorithm is proposed based on cone quasi-Newton equation.The local superlinear convergence of the algorithm is proved.This algorithm is suitable for solving unconstrained optimization problems with special structure and partially available information of Hesse matrix of objective function. The nonlinear least squares problem is a typical example of this kind of problem.A class of smoothing algorithms for solving absolute value equations are proposed, and the numerical representations of four smoothing functions are compared.The problem of absolute value equation is a kind of nondifferentiable NP-hard problem.Based on the new smoothing function, the absolute value equation is transformed into an equivalent smooth equation system, and the corresponding smoothing algorithm is used to solve the equations.Our main contribution lies in numerical experiments and comparative analysis. Through numerical comparison, we not only select the best numerical performance of the four smooth functions, but also give the ranking of the numerical performance of the four functions in terms of iteration number and computation time.
【学位授予单位】：内蒙古大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.7

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