 Solution

Since $p$ is even, differentiating $c_i^p$ givesAt a stationary point define $y_i=-c_i(x{)}^{p-1}/\mu$. Then $\nabla f(x)-Dc(x{)}^{T}y=0$, precisely the stationarity equation for the optimization Lagrangian. Bounded multipliers giveThus along any subsequence of stationary points with $x_{\mu }\to x_{\ast }$, a further subsequence of bounded multipliers converges to $y_{\ast }$, and continuity yields $c(x_{\ast })=0$ and $\nabla f(x_{\ast })-Dc(x_{\ast }{)}^T{y}_{\ast }=0$. 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.