论文标题
在优化理论中的定向渐近方法A部分A：近似，M-和混合阶平稳性
On the directional asymptotic approach in optimization theory Part A: approximate, M-, and mixed-order stationarity
论文作者
论文摘要
我们表明，对于固定的订单$γ\ geq 1 $，在欧几里得空间中，每个局部的非平滑优化问题的每个局部最小化器在经典意义上是M-Stationary（对应于订单$ 1 $）的M-Sterationary，在订单$γ$的代码构建方向上，在$γ$方向上满足平稳性条件，或者是$ undece $ $ $ $ $ $ $ $ $ $ $ $ $ $。通过排除后一种案例的约束资格不超过方向度量次要性，我们最终得到了新的必要最佳条件，其中包括限制订单$ 1 $ 1 $和$γ$的局限性工具的混合物。这些抽象的发现是针对广泛的几何约束。作为副产品，我们获得了新的约束资格，以确保当地最小化器的M阶层性。该论文通过在标准非线性，互补性约束和非线性半决赛编程的背景下说明这些结果来结束。
We show that, for a fixed order $γ\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order $1$), satisfies stationarity conditions in terms of a coderivative construction of order $γ$, or is approximately stationary with respect to a critical direction as well as $γ$ in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of order $1$ and $γ$. These abstract findings are carved out for the broad class of geometric constraints. As a byproduct, we obtain new constraint qualifications ensuring M-stationarity of local minimizers. The paper closes by illustrating these results in the context of standard nonlinear, complementarity-constrained, and nonlinear semidefinite programming.
