Power penalty multiplier limit ($y_i=-c_i^{p-1}/\mu ,|c_i|=(\mu |y_i|{)}^{1/(p-1)}$)

At a stationary point of the power penalty, define the displayed multiplier estimates. Its stationarity equation becomes $\nabla f-D{c}^{T}y=0$. If $\mu \downarrow 0$ and these multipliers are bounded, then each $c_i\to 0$. Along any convergent primal subsequence, a further multiplier subsequence converges and yields equality-constrained first-order KKT conditions at the limit. This is a statement about finite cluster points: bounded multipliers alone do not prove convergence or boundedness of the primal iterates.