多维优化问题的一个自适应两点步长算法

A self-adapted two step size Newton algorithm for multidimensional optimal problem

给出了克服牛顿算法缺陷的自适应两点步长的算法.利用拟牛顿性质得到包含前两个迭代点有关信息的迭代步长因子解析表达式,无论初始迭代点与最优解之间是否存在Hesse矩阵不正定点、鞍点和广义拐点,迭代点列自动快速逼近最优解,该算法具有自适应性且仍具有二阶收敛速度;证明了算法的收敛性,并给出了算例,利用Mathematics数学软件验证了算法的有效性.

A selfadapted two step size algorithm is presented, which overcomes the limitation of the Newton algorithm. In the first place, the analytic expression of the iteration step size factor that includes the information of two iteration points is obtained by the quasiNewton property. Whether there exists the Hesse matrix nonpositive definite point, saddle point or generalized inflexion point between the original iteration point and the best solution, the series of iteration point automatically approaches the best solution repidly. The algorithm has the selfadapted property and has a second order convergence speed. In the second place, the convergence of the algorithm is proved. One numerical example is given to illustrate the availability of the method by using Mathematics software.