  This paper mainly studies the quadratic growth and the strong metric
subregularity of the subdifferential of a function that can be represented as
the sum of a function twice differentiable in the extended sense and a
subdifferentially continuous, prox-regular, twice epi-differentiable function.
For such a function, which is not necessarily prox-regular, it is shown 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. In addition, other
characterizations of the quadratic growth and the strong metric subregularity
of the subdifferential are also given. Besides, properties of functions twice
differentiable in the extended sense are examined.
