摘要
拟牛顿方法在无约束优化中起着核心的作用．一般的拟牛顿方法是在每一步的迭代中,利用上一步产生的梯度信息,建立一个拟牛顿方程,进而求得目标函数Hes- sian阵的近似．多步拟牛顿法则是利用前m(m≥0)步的梯度信息,通过插值多项式建立一个扩展的拟牛顿方程．这两种方法的共同缺点是没有利用已知的函数值信息．本文在标准多步拟牛顿法基础上,充分利用函数值信息,构造出一个修正的带有向量参数的多步拟牛顿方程,该修正方程的多步拟牛顿法保持了较好的正定性和局部收敛性,且效率较高．数值实验也表明这个修正的算法在解决中,高维问题中比标准的多步拟牛顿方法有着更好的数值效果．
Quasi-Newton methods play a core role in unconstrained optimization. The usual quasiNewton methods, at each iteration, employ the gradients deriving from the former step and construct a quasi-Newton equation and derive the objective Hessian approximation. Multi-step methods construct a generalized quasi-Newton equation by means of interpolating polynomials and employing the gradients deriving from the previous m（m≥ 0） steps. The common drawbacks of the two methods are that they ignore the available function value information. In this paper, We construct a modified multi-step quasiNewton equation with a vector parameter which uses available function value information basing on normal multi-step quasi-Newton methods. The modified multi-step methods maintain the properties of positive-definiteness and local convergence and are more effective. Numerical experiments indicate that this modified algorithm has better numerical results than normal multi-step quasi- Newton methods in solving middle-large dimension problems.
出处
《南京大学学报（数学半年刊）》
CAS
2007年第1期142-150,共9页
Journal of Nanjing University(Mathematical Biquarterly)
关键词
无约束优化
拟牛顿方程
多步拟牛顿方法
unconstrained optimization, Quasi-Newton, Multi-step Quasi-Newton methods
