非线性约束下的梯度投影法的收敛性

Convergence of Rosen's Gradient Projection Method with Nonlinear Constraints

1960年Rosen提出的梯度投影法虽然已广泛应用得到成功,但其收敛问题20多年来一直得不到证明,同时也举不出一个反例。算法非闭是困难的原因。1986年何光中十分巧妙地证明了梯度投影法的收敛性:在n维欧氏空间中任何迭代序列的极限点均为Kuhn—Tucker点,本文将Rosen梯度投影法自然地推广到非线性约束情况,算法仍然非闭,证明了收敛性。证明的实质是从局部点态性质出发,得到一介全局收敛性的结论。

In 1960, J.B.Ronsen presented his gradient projection method for nonlinear programming with linear constraints. Although the method is successful, but the convergence of the method was a well-known difficult open problem. It was remained unsolved for more than 20 years. In 1987, He Guangzong solved the problem for the first time. In this paper an adapted method for nonlinear constraints, called 'Reduced Ronsen's Gradient Projection Method', is disscused and the convergence of the method is proved.