Exact maximum-violation penalty

An exact maximum-violation penalty replaces inequalities $g_i(x)\le 0$ by the unconstrained objective $f(x)+M\max (0,g_1(x),\dots ,g_m(x))$. If an optimal dual multiplier is ${\lambda }^{\ast }$, every $M>\Vert {\lambda }^{\ast }{\Vert }_1$ makes every penalized minimizer feasible and optimal.