自适应的信赖域方法及其工程应用研究

发布时间：2018-05-12 21:22

本文选题：机械设计 + 优化设计　；参考：《齐鲁工业大学》2012年硕士论文

【摘要】：建立优化设计问题的数学模型和选择合适的优化方法是机械优化设计两方面主要内容。为了掌握优化设计方法,需要在优化理论、建模和计算机应用等方面进行知识更新；特别是CAD/CAM以及CIMS(计算机集成制造系统)的发展,使优化设计成为当代不可缺少的技术和环节。由于机械优化设计应用数学方法来寻求机械设计的最佳方案,因此首先要根据实际的机械设计问题建立相应的数学模型,也就是说应有数学形式来描述实际设计问题。在建立数学模型时,需要应用专业知识确定设计的限制条件和所追求的目标,确立各设计变量之间的相互关系等。数学模型一旦建立,机械优化设计问题就变成一个数学求解问题。应用数学规划方法的理论,根据数学模型的特点可以选择适当的优化方法。信赖域方法是求解无约束优化问题的一类重要方法,它不要求Hessian矩阵在每个迭代点处均正定,并适合于求解一些病态问题,而且它还具有较强的收敛性和鲁棒性。由于这些优点,对信赖域方法的研究成为当今非线性优化领域内一个重要研究方向。本文首先研究了一类自适应的信赖域算法：具体给出了算法模型；在合理的假设条件下,对它的全局收敛性和超线性收敛性进行了论证；并把它应用于机械领域。本论文的结构如下： 1.给出了一类新的信赖域方法。首先在非单调技巧的帮助下,构造了一类自调节信赖域半径的方法。在算法的实现的过程中,不要求函数值在每一步都下降,特别是对于目标函数存在弯曲峡谷的情形,非单调性能加快算法的收敛速度。在每个迭代点处,我们充分利用当前迭代点和先前迭代点的信息来构造信赖域半径,使得二次函数模型和目标函数在当前信赖域内具有更好的一致性。这种自适应的方法不但克服了初始信赖域半径选取的盲目性,而且降低了问题的复杂性,使算法的速度得以提高；最后在某些假设下,对所给的算法收敛性质进行了证明。 2.在二次模型信赖域子问题的为基础,研究了一种带线搜索的非单调信赖域方法。在当前迭代点处,我们利用信赖域技巧来寻找下一个迭代点,若子问题的近似解不能被接受,使用非单调线搜索寻找下一个迭代点。于是在当前迭代点处,仅仅需要求解一次子问题就可以找到下一个成功的迭代点,这样可以避免过大的计算量,加快算法的收敛速度。在合适的条件下,证明了这种算法是全局收敛的。 3.通过实例给出了机械优化设计的一般步骤,并研究了非单调自适应的信赖域方法在机械优化设计中的应用,对该方法和存在的优化方法进行了比较,从而表明非单调自适应的信赖域方法对于解决机械优化问题是有效的和可行的。
[Abstract]:The establishment of mathematical model and the selection of appropriate optimization methods are two main contents of mechanical optimization design. In order to master the optimization design method, it is necessary to update the knowledge in optimization theory, modeling and computer application, especially the development of CAD/CAM and CIMS (computer Integrated Manufacturing system). Make the optimization design become the contemporary indispensable technology and link. Due to the application of mathematical method to seek the best scheme of mechanical design, it is necessary to establish the corresponding mathematical model according to the actual mechanical design problem, that is to say, there should be mathematical form to describe the practical design problem. In the process of establishing mathematical model, we need to apply professional knowledge to determine the limit condition and target of design, and to establish the relationship between design variables and so on. Once the mathematical model is established, the mechanical optimization design problem becomes a mathematical solution problem. Based on the theory of mathematical programming, the appropriate optimization method can be selected according to the characteristics of mathematical model. Trust region method is an important method for solving unconstrained optimization problems. It does not require the Hessian matrix to be positive definite at every iteration point, and is suitable for solving some ill-conditioned problems. Moreover, it has strong convergence and robustness. Because of these advantages, the research of trust region method has become an important research direction in the field of nonlinear optimization. In this paper, we first study a kind of adaptive trust region algorithm: we give the algorithm model in detail, prove its global convergence and superlinear convergence under reasonable assumptions, and apply it to the mechanical field. The structure of this thesis is as follows: 1. A new trust region method is given. Firstly, with the help of nonmonotone technique, a method of self adjusting trust region radius is constructed. In the implementation of the algorithm, the value of the function is not required to decrease at every step, especially in the case of the objective function with curved canyons, the non-monotonicity can speed up the convergence of the algorithm. At each iteration point, we make full use of the information of the current iteration point and the previous iteration point to construct the trust region radius, so that the quadratic function model and the objective function are more consistent in the current trust region. This adaptive method not only overcomes the blindness of selecting the initial trust region radius, but also reduces the complexity of the problem and improves the speed of the algorithm. Finally, under some assumptions, the convergence property of the proposed algorithm is proved. 2. Based on the trust region subproblem of quadratic model, a nonmonotone trust region method with line search is studied. At the current iteration point, we use the trust region technique to find the next iteration point. If the approximate solution of the subproblem is not acceptable, we use the non-monotone line search to find the next iteration point. Therefore, at the current iteration point, the next successful iteration point can be found only by solving one subproblem, which can avoid too much computation and speed up the convergence of the algorithm. Under suitable conditions, it is proved that the algorithm is globally convergent. 3. The general steps of mechanical optimization design are given through an example, and the application of nonmonotone adaptive trust region method in mechanical optimization design is studied, and the comparison between this method and the existing optimization method is given. It is shown that the nonmonotone adaptive trust region method is effective and feasible for solving mechanical optimization problems.
【学位授予单位】：齐鲁工业大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：TH122

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