自适应共轭梯度法的研究

发布时间：2018-04-22 18:29

本文选题：共轭梯度法 + 自适应共轭条件　；参考：《西安电子科技大学》2015年博士论文

【摘要】：无约束优化问题广泛应用于经济计划、工程设计、生产管理、国防与航空航天等重要领域,因此构造大规模优化问题的计算方法,研究这些方法的理论性质及其实际数值表现具有重要的理论意义和实际应用价值.存储量小和迭代简单的特点使共轭梯度法在求解大规模问题的算法中脱颖而出.在过去的20年中,充分下降条件和共轭性使得共轭梯度法在优化领域更为活跃.本文在总结己有非线性共轭梯度算法的基础上,从实用角度出发,设计出若干能满足上述两个条件的白适应共轭梯度法.主要具体工作如下：1.我们引入了两类白适应的共轭梯度法,该法在每步迭代可满足充分下降条件.与现有方法不同的是,本文提出新的共轭条件是动态调整的,它可视作HS共轭性和DL共轭性的继承与发展.在适当情况下,可证明该法对一般函数全局收敛.2.我们对六类基本共轭梯度法进行了修正,其中的搜索方向满足不依赖于任何搜索条件的充分下降条件.此外,我们提出了一个一般形式的共轭梯度法,对应的搜索方向总是充分下降方向.该方法无须Yuan提出的”步长要有正的下界”的假设条件,可以建立算法的全局收敛性.3.我们构造了一个非一致凸的二维函数,它可以说明这样一种可能性,即无论TTCG方法在极小化我们提出的函数时是否收敛,TTCG方法的收敛性分析中关于‘sTKyKτ(τ0是常数)”这一充分条件都不成立.主要原因在于在数量上,sTkyk是恢||2的高阶无穷小.此外,我们提出了一类具有一般形式的三项共轭梯度法,它的搜索方向同时满足白适应共轭条件和充分下降条件.4.原TTDES方法中的某些结果因为参数选取不当需要修正,我们在迭代矩阵条件数最小的意义下找至TTDES方法的最优参数.具体地,既然该迭代矩阵既非对称也不正则,在讨论条件数时,一种谨慎而合理的策略是采取奇异值分析而非特征值分析.5.我们通过不同搜索方向之间的仿射组合而得到新的Hestenes-Stiefel类型不Polak-Ribiere-Polyak类型的三项共轭梯度法.在迭代过程中,搜索方向满足充分下降条件,并能接近拟牛顿方向或满足共轭条件.算法在Wolfe搜索下收敛.6.数值结果显示,本文提出的上述方法适于求解大型优化问题,从而是有效的.白适应的算法机制不仅有益于共轭梯度法的理论与计算,随着时间的推移,它将展示出更有意义的重要性.
[Abstract]:Unconstrained optimization problems are widely used in important fields such as economic planning, engineering design, production management, national defense and aerospace. It is of great theoretical significance and practical application value to study the theoretical properties and practical numerical expression of these methods. The characteristics of small storage and simple iteration make the conjugate gradient method stand out in the algorithm for solving large-scale problems. In the past 20 years, the sufficient descent condition and conjugacy make the conjugate gradient method more active in the optimization field. On the basis of summarizing the existing nonlinear conjugate gradient algorithm and from the practical point of view, this paper designs some white adaptive conjugate gradient methods which can satisfy the above two conditions. The main specific work is as follows: 1. We introduce two classes of white adaptive conjugate gradient method, which can satisfy the sufficient descent condition at each step iteration. Different from the existing methods, the new conjugate condition is dynamically adjusted, which can be regarded as the inheritance and development of HS conjugacy and DL conjugacy. It is proved that the global convergence of the method for general functions. 2. We modify six basic conjugate gradient methods in which the search direction satisfies the sufficient descent condition which does not depend on any search conditions. In addition, we propose a general form of conjugate gradient method, the corresponding search direction is always sufficient descent direction. This method does not need the assumption that the step size should have a positive lower bound, and the global convergence of the algorithm can be established by using the assumption of "step size must have positive lower bound" proposed by Yuan. We construct a nonuniformly convex two-dimensional function that illustrates the possibility, That is, no matter whether the TTCG method converges or not in the minimization of the functions proposed by us, the sufficient condition that the TKK 蟿 (蟿 0 is a constant) in the analysis of the convergence of the TTCG method does not hold true. The main reason is that s Tkyk is the higher order infinitesimal of Hui 2 in quantity. In addition, we propose a class of three-term conjugate gradient method with general form, whose search direction satisfies both the white adaptive conjugate condition and the sufficient descent condition .4. Some results in the original TTDES method need to be corrected because of improper selection of parameters. We find the optimal parameters of the TTDES method in the sense of the minimum number of conditions of the iterative matrix. Specifically, since the iterative matrix is neither asymmetric nor regular, a prudent and reasonable strategy for discussing condition numbers is to adopt singular value analysis rather than eigenvalue analysis .5. By affine combinations between different search directions, we obtain a new three-term conjugate gradient method of Hestenes-Stiefel type non- type. In the iterative process the search direction satisfies the sufficient descent condition and can approach the quasi-Newtonian direction or satisfy the conjugate condition. The algorithm converges. 6 under Wolfe search. Numerical results show that the proposed method is suitable for solving large scale optimization problems. The algorithm mechanism of white adaptation is not only beneficial to the theory and calculation of conjugate gradient method, but also shows more significance over time.
【学位授予单位】：西安电子科技大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O224

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