论文标题
值函数的定向子差异
Directional subdifferential of the value function
论文作者
论文摘要
值函数的定向子差异可以估计在特定方向扰动下最佳值变化的程度。在本文中，我们得出了一个非常通用的参数优化问题的定向限制和奇异次数的估计值。我们获得了在定向子差异方面的局部半连续函数的定向Lipschitz度的表征。基于此特征和定向单数亚不同差异的派生的上限估计，我们能够为该值函数的定向Lipschitz获得足够的条件。最后，当涉及的所有功能平滑，扰动是加性，约束独立于参数独立于参数或约束是平等性和不平等时，我们为各种情况指定了这些结果。我们的结果扩展了对值函数灵敏度的相应结果，以允许定向扰动。即使在完全扰动的情况下，我们的结果恢复甚至扩展了一些现有结果，包括Danskin定理。
The directional subdifferential of the value function gives an estimate on how much the optimal value changes under a perturbation in a certain direction. In this paper we derive upper estimates for the directional limiting and singular subdifferential of the value function for a very general parametric optimization problem. We obtain a characterization for the directional Lipschitzness of a locally lower semicontinuous function in terms of the directional subdifferentials. Based on this characterization and the derived upper estimate for the directional singular subdifferential, we are able to obtain a sufficient condition for the directional Lipschitzness of the value function. Finally, we specify these results for various cases when all functions involved are smooth, when the perturbation is additive, when the constraint is independent of the parameter, or when the constraints are equalities and inequalities. Our results extend the corresponding results on the sensitivity of the value function to allow directional perturbations. Even in the case of full perturbations, our results recover or even extend some existing results, including the Danskin's theorem.
