摘要
已证明,当最速下降法的步长为系数矩阵特征值的倒数时,任意非奇异矩阵都可以在m步内收敛到精确解,这里m为系数矩阵最小多项式的次数。这是一种理想化的最速下降法。由于特征值的计算并不容易,因此只能用其近似估算值代替。分析了近似特征值获取方法并研究了其误差对迭代的影响,从而给出了逼近理想化的最速下降法的一般方法。作为一个例子,给出了一种高效的自适应循环最速下降法:每当求出最优步长h后,将算法变成定步长最速下降法并用该步长重复M步,当目标函数或梯度模反而变大时则放弃重复。这里,M可根据经验预先确定。该算法保证了目标函数值的单调下降性质。将上述结果推广至一般函数的无约束最优化,并对一些典型测试函数的计算表明:该算法的收敛速度优于共轭方法和变尺度法,内存需求则与共轭方法相当。
It was proved that when the step of steepest descent method is the reciprocals of the eigenvalues of the non-singular coefficient matrix A of equation Ax=b,the iteration can converge to exact solution in m-step,where m is its degree of minimal polynomial.This is an idealized steepest descent method.However,since the calculation of eigenvalues is not easy,only approximate estimates can be used in actuarial calculation.This paper studies the method to obtained the approximate eigenvalue and the impact of its error,which gives general approximation to idealized steepest descent method.An efficient steepest descent method was given as an example：As soon as a step length h is obtained by a line search technique,the length h was used M times during following iterations if the objective function or the norm of gradient decreases.The results were applied to the unconstrained optimization for general function. The calculation shows that the rate of convergence of this work is faster than that of conjugate method and DFP method but its memory consumption is only similar to that of conjugate method.
出处
《上海第二工业大学学报》
2011年第1期8-13,共6页
Journal of Shanghai Polytechnic University
关键词
最速下降法
特征值
最小多项式
无约束优化
steepest descent method
characteristic values
minimal polynomial
unconstrained optimization
