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Quadratic Growth and Strong Metric Subregularity of the Subdifferential via Subgradient Graphical Derivative
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.
Read More: https://epubs.siam.org/doi/abs/10.1137/19M1242732
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