  This paper is devoted to the study of second order optimality conditions for
strong local minimizers in the frameworks of unconstrained and constrained
optimization problems in finite dimensions via 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 quadratic growth
condition. It is also a necessary condition for the latter property when the
function is either subdifferentially continuous, prox-regular, twice
epi-differentiable or variationally convex. By applying our results to the
$\mathcal{C}^2$-cone reducible constrained programs, we establish no-gap second
order optimality conditions for (strong) local minimizers under the metric
subregularity constraint qualification. These results extend the classical
second order optimality conditions by surpassing the well-known Robinson's
constraint qualification. Our approach also highlights the interconnection
between the strong metric subregularity of subdifferential and quadratic growth
condition in optimization problems.
