Slater Condition for Tangent Derivatives
Slater Condition for Tangent Derivatives
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DOI：
10.1287/moor.2021.1246
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发表时间：
2022-02
期刊：
影响因子：
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通讯作者：
Xi Yin Zheng
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文献类型：
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作者：
Xi Yin Zheng
注意到现有的Slater条件作为优化中的基本约束条件仅适用于凸设置，我们引入并研究了一般向量值函数F关于闭凸锥K的Bouligand和Clarke正切导数的Slater条件。在没有任何假设的情况下，证明了当目标函数F经历小Lipschitz（平静）时，Clarke（分别为Bouligand）关于K的正切导数的Slater条件总是稳定的扰动。据此，我们证明，如果F的Clarke（Bouligand）正切导数满足Slater条件（相对于K），则当F经历小Lipschitz（平静正则）扰动时，由F确定的二次曲线不等式具有稳定的度量次正则性。在复合凸的情况下，逆推论也被证明是正确的。此外，在F的正切导数的Slater条件下，证明了F的子水平集的法锥可以通过F的次微分来表示，从而改善了光滑或凸情况下的相应结果。作为应用，在没有任何限定假设的情况下，我们改进并推广了 Cabot 和 Thibault 提出的法向圆锥到凸子水平集的公式 [(2014)，法向圆锥到子水平集的序列公式。美国数学会汇刊 366(12):6591–6628]。借助这些公式，建立了一些新的卡鲁什-库恩-塔克最优条件。
Noting that the existing Slater condition, as a fundamental constraint qualification in optimization, is only applicable in the convex setting, we introduce and study the Slater condition for the Bouligand and Clarke tangent derivatives of a general vector-valued function F with respect to a closed convex cone K. Without any assumption, it is proved that the Slater condition for the Clarke (respectively, Bouligand) tangent derivative with respect to K is always stable when the objective function F undergoes small Lipschitz (calm) perturbations. Based on this, we prove that if the Clarke (Bouligand) tangent derivative of F satisfies the Slater condition (with respect to K) then the conic inequality determined by F has a stable metric subregularity when F undergoes small Lipschitz (calm regular) perturbations. In the composite-convexity case, the converse implication is also proved to be true. Moreover, under the Slater condition for the tangent derivative of F, it is proved that the normal cone to the sublevel set of F can be formulated by the subdifferential of F, which improves the corresponding results in either the smooth or convex case. As applications, without any qualification assumption, we improve and generalize formulas for the normal cone to a convex sublevel set by Cabot and Thibault [(2014), Sequential formulae for the normal cone to sublevel sets. Transactions of the American Mathematical Society 366(12):6591–6628]. With the help of these formulas, some new Karush–Kuhn–Tucker optimality conditions are established.