SIAM Journal on Optimization, Volume 31, Issue 1, Page 545-568, January 2021.
This paper mainly studies the relationship between quadratic growth and strong metric subregularity of the subdifferential in finite dimensional settings by using the subgradient graphical derivative. We prove that the positive definiteness of the subgradient graphical derivative of an extended-real-valued lower semicontinuous proper function at a proximal stationary point is sufficient for the point to be a local minimizer at which the subdifferential is strongly subregular for $0.$ The latter was known to imply the quadratic growth. When the function is either subdifferentially continuous, prox-regular, twice epidifferentiable, or variationally convex, we show that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. For $\mathcal{C}^2$-cone reducible constrained programs satisfying the metric subregularity constraint qualification, we obtain the same results for the sum of the objective function and the indicator function of the feasible set.

中文翻译：

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通过次梯度图形导数的二次微分的二次增长和强度量次正则性

SIAM优化杂志，第31卷，第1期，第545-568页，2021年1月。
本文主要通过使用次梯度图形导数研究在有限维设置中二次增长与次微分的强度量次正则性之间的关系。我们证明了扩展的实值下半连续固有函数在近端固定点的次梯度图形导数的正定性足以使该点成为局部最小化子，在该子点上，微分的强次正则性为$ 0。众所周知，它暗示着二次增长。当函数要么是亚微分连续的，近似正则的，两次表微分的或凸变的，我们就表明二次增长，即局部极小处的亚微分的强度量次正则性，次梯度图形导数在固定点的正定性是等价的。对于满足度量次规则约束条件的$ \ mathcal {C} ^ 2 $ -cone可约约束程序，对于可行集的目标函数和指标函数之和，我们获得相同的结果。