 Solution

The equation $F_{\mu }(x,y)=0$, together with $c(x)>0$ and $y>0$, is the perturbed Karush-Kuhn-Tucker conditions: stationarity is unchanged, while exact complementary slackness becomes $c_i(x)y_i=\mu$ for each $i$. A finite limit of such points as $\mu \downarrow 0$ satisfies the original conditions by continuity.

For a strictly feasible $x$, the second block uniquely specifies $y_i=\mu /c_i(x)$. Substitution into the first block gives precisely $\nabla B_{\mu }(x)=0$. Thuswithin the stated strictly positive domain. These are stationary barrier points; they need not all be local minimizers in a nonconvex problem.