The scaled KKT residual vanishes exactly at KKT tuples

If the residual is zero, each summand in [F1] is nonnegative and their sum is zero, so each norm is zero. The stationarity and equality terms then give stationarity and $h(x)=0$. The positive-part term is zero exactly when $g_i(x)\le 0$ for every $i$, and the negative-multiplier term is zero exactly when ${\lambda }_i\ge 0$ for every $i$. The last term gives ${\lambda }_i{g}_i(x)=0$ coordinatewise. These are all conditions in [F2].

Conversely, if the KKT conditions hold, every vector inside each norm in [F1] is zero: feasibility makes $h=0$ and $[g{]}^{+}=0$, dual feasibility makes $[-\lambda {]}^{+}=0$, complementarity makes $\lambda \circ g=0$, and stationarity makes the first vector zero. Thus the residual is zero.