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Convert to minimization with $f=-F$ and retain $g\ge 0$. A continuously differentiable inequality augmented Lagrangian method uses the merit functionFor $g_i\le \mu {\lambda }_i$ its contribution is $-{\lambda }_i{g}_i+g_i^2/(2\mu )$; otherwise it is the constant $-\mu {\lambda }_i^2/2$. This shifted truncation permits multipliers to decrease as well as increase while remaining nonnegative.

A useful initial estimate comes from the preceding pure quadratic penalty method at ${\mu }_{-1}$:Here $g(-1,-1)=(3,18,-2)$, so with ${\mu }_{-1}=2$ chooseThen use this estimate and ${\mu }_0=1$ in the first augmented subproblem. Initial estimates are a choice, rather than uniquely specified data; zero multipliers would also define an initialization.

For constraints $h=0$, $g\ge 0$, minimize the displayed merit function for progressively smaller $\mu >0$, where $g_i^{-}=\min (g_i,0)$. Increasing the quadratic penalty discourages constraint violation. In a violated inequality, differentiation gives the multiplier estimate ${\lambda }_i=-g_i/\mu$; an inactive inequality has estimate zero. Thus obtaining small violations by this method alone typically requires a very small $\mu$, which may cause a large condition number in the inner minimization.