基于二次多项式的积极集光滑化max函数及其在无约束minimax问题中的应用

ACTIVE SET SMOOTHING MAX FUNCTION BASED ON QUADRATIC POLYNOMIAL AND ITS APPLICATION TO UNCONSTRAINED MINIMAX PROBLEMS

本文基于分段二次多项式方程,构造了一种积极集策略的光滑化max函数.通过给出与光滑化max函数相关的分量函数指标集的直接计算方法,将分段二次多项式方程转化为一般二次多项式方程.利用二次多项式方程根的性质,给出了该光滑化max函数的稳定计算策略,证明了其具有一阶光滑性,其梯度函数具有局部Lipschitz连续性和强半光滑性.该光滑化max函数仅与函数值较大的分量函数相关,适用于含分量函数较多且复杂的max函数的问题.为了验证其效率,本文基于该函数构造了一种解含多个复杂分量函数的无约束minimax问题的光滑化算法,数值实验表明了该光滑化max函数的可行性及有效性.

In this paper,based on the piecewise quadratic polynomial equation,a smoothing max function with the active set strategy is proposed.The piecewise quadratic polynomial equation is transformed to a general quadratic polynomial equation by calculating the index set related to the smoothing max function.Then,by the properties of roots for the quadratic polynomial equation,a stable calculation strategy for the smoothing max function is given.The smoothing max function is continuously differentiable,its gradient function is locally Lipschitz continuous and strongly semi-smooth.It is only related to the component functions with large function value,hence,it is suitable for problems containing max function defined by a large number of complex component functions.In order to show the efficiency of the smoothing max function,a smoothing method based on the smoothing max function is proposed for unconstrained minimax problems with multiple complex component functions.Preliminary numerical experiments show the feasibility and efficiency of the smoothing max function.