= Solution
Since $p$ is even, differentiating $c_i^p$ gives
$$
\boxed{\nabla_x\Phi(x,\mu)=\nabla f(x)+\frac1\mu\sum_i c_i(x)^{p-1}\nabla c_i(x).}
$$
At a stationary point define $y_i=-c_i(x)^{p-1}/\mu$. Then $\nabla f(x)-Dc(x)^Ty=0$, precisely the stationarity equation for the <optimization Lagrangian>. Bounded multipliers give
$$
\boxed{|c_i(x)|=(\mu|y_i|)^{1/(p-1)}\longrightarrow0.}
$$
Thus along any subsequence of stationary points with $x_\mu\to x_*$, a further subsequence of bounded multipliers converges to $y_*$, and continuity yields $c(x_*)=0$ and $\nabla f(x_*)-Dc(x_*)^Ty_*=0$. These are the equality-constrained first-order conditions. Bounded multipliers alone do not establish convergence of the primal points; the precise conclusion concerns their finite cluster points. This is the <power penalty multiplier limit>.
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