一个求解二次无约束大型病态问题的混合-下降算法

从最速下降法在求解病态优化问题时常会出现"锯齿"现象,且所得解严重失真这一问题出发,利用微分方程数值积分求解所建立的常微分方程自治系统初值问题得到的解作为最速下降法的经过改进的可接收初始点,并与最速下降法结合得到一个求解二次无约束病态问题的混合-下降算法.算法具有全局收敛性.初步数值实验表明,将算法运用于1 000阶Hilbert矩阵所构成的二次无约束大型病态问题,能够求得具有3位有效数字的解,说明新算法具有良好的稳定性及较强的抗病态能力.

A mixed-degressive algorithm for large-scale ill-conditioned quadratic unconstrained optimization

In solving ill-conditioned unconstrained optimization the steepest descent method(SDM)often raises the zigzaging phenomenon and results in slow convergence and completed wrong solutions.Provided a acceptabil initial point for SDM.Then a mixed-degressive algorithm for large-scale ill-conditioned quadratic unconstrained optimization was put forward by combining numerical integration with SDM.It is global convergent.The preliminary numerical results obtained by executing new algorithm on test problems are presented.Numerical tests showed excellent accuracy and stability of the new algorithm.Test problem with a matrix of Hilbert of 1 000 order as its Hessian matrix,the result has 3 effective digitals.All these showed that the proposed algorithm have expected effective anti-ill-conditioned ability.