论文标题
为什么二阶足够条件在某种程度上很容易或 - 重新访问第二次下属的微积分
Why second-order sufficient conditions are, in a way, easy -- or -- revisiting calculus for second subderivatives
论文作者
论文摘要
在本文中，我们阅读了二阶足够最佳条件的经典主题，以优化非平滑结构的优化问题。基于目标函数的所谓第二次亚衍生物和与可行集合相关的指标函数的次数，人们很容易获得抽象形式的二阶足够最佳条件。为了利用问题的进一步结构，例如，在光滑转换下封闭的封闭集的前图中给出的可行术语中的综合项，以使这些条件完全明确，我们研究了在轻度条件下第二次下属的计算规则。确切地说，我们研究了链条规则和边缘函数规则，然后分别给出了前图像和图像规则。事实证明，链条规则和前图规则得出的估计值较低，以便获得足够的最佳条件。在相对温和的内部平静*假设下，对边际功能和图像规则的相似估计是有效的。我们的发现通过了几个示例，包括复合，分离和非线性二阶编程的问题。
In this paper, we readdress the classical topic of second-order sufficient optimality conditions for optimization problems with nonsmooth structure. Based on the so-called second subderivative of the objective function and of the indicator function associated with the feasible set, one easily obtains second-order sufficient optimality conditions of abstract form. In order to exploit further structure of the problem, e.g., composite terms in the objective function or feasible sets given as (images of) pre-images of closed sets under smooth transformations, to make these conditions fully explicit, we study calculus rules for the second subderivative under mild conditions. To be precise, we investigate a chain rule and a marginal function rule, which then also give a pre-image and image rule, respectively. As it turns out, the chain rule and the pre-image rule yield lower estimates desirable in order to obtain sufficient optimality conditions for free. Similar estimates for the marginal function and the image rule are valid under a comparatively mild inner calmness* assumption. Our findings are illustrated by several examples including problems from composite, disjunctive, and nonlinear second-order cone programming.
