极大值函数的一类光滑逼近函数的性质研究

Some Properties of a Smoothing Approximation for Maximum Function

针对极大值函数的一类光滑逼近——凝聚函数,对其作进一步研究.指出凝聚函数的一阶导数对光滑参数取极限时恰好得到极大值函数的一个次梯度,从而凝聚函数不仅可以一致逼近极大值函数,而且该函数富含极大值函数的一阶信息,可很好的刻画极大值函数的一阶特征.进一步,对光滑逼近函数的光滑参数做简单分析,得到的结果揭示了光滑参数的变动对凝聚函数的影响.并分别以正值函数及绝对值函数为例,对所得到的结果给出几何说明.

This paper is a further exploration of the approximation scheme that is called aggregate function for maximum function. One subgradient of the maximum function can be obtained by computing classical gradients of aggregate function with the smoothing parameter goes to infinity. The aggregate function provides a wealth of first order information on the behavior of maximum function. For the analysis of aggregate function as a function of the smoothing parameter, we analyze the influence of the smoothing parameter to the aggregate function.