求解无约束优化问题的两种修正共轭梯度法及其应用

Abd Elhamid Mehamdia, Y. Chaib, T. Bechouat
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引用次数: 0

摘要

共轭梯度法是求解线性方程组和非线性优化问题的一类常用迭代方法,因为它不需要存储任何矩阵。为了获得一种理论有效和数值有效的方法,提出了两种修正共轭梯度法(MCB1和MCB2方法)。其中,两种方法的系数βk是受现有共轭梯度方法中共轭梯度参数结构的启发。在强Wolfe线搜索下,证明了MCB1方法的充分下降性和全局收敛性。此外,MCB2方法产生的下降方向独立于任何线搜索,并且在使用强Wolfe线搜索时具有良好的收敛性。初步的数值结果表明,MCB1和MCB2方法在最小化一些无约束优化问题上是有效的和鲁棒的,并且每种修正方法都优于四种著名的共轭梯度方法。在此基础上,将所提算法扩展到求解模态函数问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Two modified conjugate gradient methods for solving unconstrained optimization and application
Conjugate gradient methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems as they do not require the storage of any matrices. In order to obtain a theoretically effective and numerically efficient method, two modified conjugate gradient methods ( called the MCB1 and MCB2 methods ) are proposed. In which the coefficient βk in the two proposed methods is inspired by the structure of the conjugate gradient parameters in some existing conjugate gradient methods. Under the strong Wolfe line search, the sufficient descent property and global convergence of the MCB1 method are proved. Moreover, the MCB2 method generates a descent direction independently of any line search and produces good convergence properties when the strong Wolfe line search is employed. Preliminary numerical results show that the MCB1 and MCB2 methods are effective and robust in minimizing some unconstrained optimization problems and each of these modifications outperforms the four famous conjugate gradient methods. Furthermore, the proposed algorithms were extended to solve the problem of mode function.
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