基于微分方程模型的不定信赖域算法研究

发布时间：2018-06-30 18:02

本文选题：不定矩阵 + 信赖域子问题　；参考：《太原科技大学》2017年硕士论文

【摘要】：信赖域算法以其较强的适定性以及全局收敛性受到最优化界许多研究者的关注。作为一类求解无约束优化问题的重要数值计算方法,其成为非线性规划问题的研究热点。信赖域子问题的构造和有效求解是实现信赖域算法的关键。目前已经建立了多种信赖域子问题的模型,如二次模型、锥模型、新锥模型以及张量模型等等。由于用二次函数模型去逼近目标函数计算方便且形式简单,因此二次函数模型是一种最基础、最重要的模型。二次函数模型信赖域子问题是目前信赖域研究的热点之一。微分方程模型是近年来求解二次函数模型信赖域子问题的一种有效的新型算法。目前大多算法是围绕正定信赖域子问题进行研究,不定信赖域子问题研究较少,本文主要围绕不定信赖域子问题研究。求解不定信赖域子问题的关键是通过修正不定矩阵将不定信赖域子问题转化为正定信赖域子问题。本文利用Bunch-Parlett分解、修改Cholesky分解方法修正不定矩阵,对求解微分方程模型的Adams四阶方法、Heun三阶方法和分段三次Hermite插值法、显式欧拉法等四种算法进行修正,提出了四种求解基于微分方程模型的不定信赖域子问题算法。数值实验结果表明:这四种算法优于其他不定算法,且更有效。此外,本文从理论上证明了修正休恩三阶算法的适定性。
[Abstract]:Because of its strong fitness and global convergence, trust region algorithm has attracted many researchers' attention. As an important numerical method for solving unconstrained optimization problems, it has become a research hotspot in nonlinear programming problems. The construction and effective solution of the trust region subproblem is the key to realize the trust region algorithm. Several models of trust region subproblems have been established, such as quadratic model, cone model, new cone model, Zhang Liang model and so on. The quadratic function model is the most basic and important model because it is convenient and simple to calculate the objective function by using the quadratic function model. The trust region subproblem of quadratic function model is one of the hotspots in the research of trust region. Differential equation model is an effective new algorithm for solving trust region subproblem of quadratic function model in recent years. At present, most of the algorithms focus on the positive definite trust region subproblem, but less on the indefinite trust region subproblem, and this paper mainly focuses on the indefinite trust region subproblem. The key to solve the indefinite trust region subproblem is to transform the indefinite trust region subproblem into a positive definite trust region subproblem by modifying the indefinite matrix. In this paper, by using Bunch-Parlett decomposition, modified Cholesky decomposition method is used to modify the indeterminate matrix, and the Adams fourth-order method Heun third-order method, piecewise cubic Hermite interpolation method and explicit Euler method are modified. In this paper, four subproblems based on the model of differential equations are proposed. The numerical results show that these four algorithms are better than other uncertain algorithms and are more effective. In addition, this paper theoretically proves the fitness of modified Hugh's third order algorithm.
【学位授予单位】：太原科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O221.2

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