极小化r个最大函数和的一阶光滑化方法

First-order smoothing method for minimizing the sum of the r-largest functions

研究如下问题:已给m个定义在n维欧几里?空间的函数,在这m个函数中求r个最大值函数的最小值,其中1≤r≤m.显然,该问题是非光滑最优化问题,不能直接用一阶最优化方法或梯度法来求解.将此问题转化为只包含最大值函数m ax{0,t}的非光滑问题,对该非光滑问题提出了一种收敛的一阶光滑化算法.

To a collection of m functions defined on R^n, the minimal value of the r （ 1 ≤ r ≤ m） largest functions is need to be found . It is obvious that this problem is nonsmooth optimization problem, and it cannot be solved using any first-order or gradient unconstrained minimization algorithms. In this paper, the problem is reformulated as a nonsmooth problem that involves only the maximum function max {0, t}. In order to solve nonsmooth problem, a new convergent first-order smoothing method is developed.