Power penalty
= Power penalty
{title2=$\Phi(x,\mu)=f(x)+\|c(x)\|_p^p/(p\mu)$}
For an equality-constrained optimization problem, an even integer $p\geq2$ makes this a nonnegative penalty for constraint violation. Its derivative is $\nabla f+\mu^{-1}Dc^Tc^{p-1}$, with the power applied coordinatewise. The case $p=2$ is the quadratic penalty. Reducing $\mu$ increases the cost of infeasibility; it does not itself ensure that computed stationary points or minimizers stay bounded.