Solution map of a parametric optimization problem ($S(a)={\mathrm{argmin}}_x{F}_a(x)$)

The solution map sends a data parameter $a$ to the set of minimizers of the associated objective, $S(a)={\mathrm{argmin}}_x{F}_a(x)$. Existence, uniqueness and local stability are distinct questions. Strong convexity gives uniqueness when a minimizer exists; the Aubin property gives a perturbation bound.