Augmented Lagrangian method (${\lambda }_i^{+}=\max (0,{\lambda }_i-g_i/\mu )$)

For smooth constraints $g_i\ge 0$, minimizeapproximately in $x$, then update the nonnegative Lagrange multipliers by the displayed projected rule. Equality constraints instead contribute $-{\nu }_j{h}_j+h_j^2/(2\mu )$ and use the unprojected update ${\nu }_j^{+}={\nu }_j-h_j/\mu$. The construction combines Lagrangian duality with a quadratic penalty method.

For an inequality, the exact identity is $\min (g_i,\mu {\lambda }_i)=\mu ({\lambda }_i-{\lambda }_i^{+})$. Its unshifted negative part satisfies $\min (g_i,0)=\mu \min ({\lambda }_i-{\lambda }_i^{+},0)$. On a violated constraint, accurate new multipliers therefore give violation of the order of $\mu$ times the multiplier error. This can achieve high feasibility accuracy without a vanishingly small $\mu$, reducing condition number difficulties. The negative-part qualification is essential if an iterate approaches a binding constraint from the feasible side.