随机规划的弱微分性

WEAK DIFFERENTIABILITY OF STOCHASTIC PROGRAMMING

本文将随机函数ν（ｘ，ｗ）引入随机规划问题ｚ（ν（ｗ））＝ｓｕｐｙ∈Ｙ｛Ｅｆ（ν（ｗ），ｙ）｜Ｅｇｊ（ν（ｗ），ｙ）０，ｊ＝１，Ｊ｝中．对相应的最优化问题的稳定性和最优值函数的可微性作了一些探讨．并得出了ｚ（ν（ｗ））的弱可微的充分性条件及相应的微分表达式．

The general form of stochastic programming problem is: z(ν(w))= sup y∈Y{ E f(ν(w),y)| E g j(ν(w),y)0,j= 1,J },(1) where ν(w) is a random variable or a random vector. In this paper, we bring random function ν(x,w) into stochastic programming problem, analyse the stability of the corresponding optimization problems and study the directional derivatives of optimal value functions. Based on it, we derive the weak differentiability theory of the optimal value functions. These results provide the theoretical base for constructing approximate algorithms, especially, for estimating approximation errors and improving approximate values.