一类求解极小极大问题的算法

A ε-algorithm for solving minmax problem

无约束非线性极小极大问题是最优化数值计算领域中十分活跃的研究课题之一,因此,对于无约束非线性极小极大问题,如何设计快速有效的算法一直都是优化工作者十分关心的问题。文中介绍了无约束非线性极小极大问题算法的研究意义及应用领域,分析了现有极小极大问题算法的研究现状,针对极大值函数的特性,给出了极大值函数的次梯度与ε次梯度之间及极大值函数的次梯度的凸锥与次梯度之间的一种包含关系,得到了计算极大值函数的ε次梯度的数值方法,从而构造出了一种求解极小极大问题的ε-算法,并且证明了算法的收敛性,初步的数值例子表明算法是有效的,且具有大范围收敛的特点。

Numerical methods for unconstrained nonlinear minimax optimization is an active subject in numerical analysis. Therefore, how to design fast and effective algorithms for unconstrained optimization is an important problem. This paper introduces the unconstrained nonlinear minimax optimization theory and algorithm, including history of the development, research signification, application areas and study status quo. According to the characteristics of maximal function, given the containment relationship be tween subgradient and c subgradient of maximal function and the convex cone of subgradient and sub gradient of maximal function, got the numerical methods in calculating subgradient of maximal func tion, accordingly constructing e algorithm in solving minimax problem and proving the convergence of the algorithm. The preliminary numerical example shows that the algorithm is effective, with a large of convergence characteristics.