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A merit function is a scalar objective function used by an iterative constrained algorithm to balance objective reduction and feasibility. It permits a line search or other acceptance test even when the original objective function alone is insufficient to judge progress. Quadratic penalty methods and the augmented Lagrangian method give two constructions with different treatments of multiplier information.

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.

Complete the square in the augmented Lagrangian method objective:For fixed $u,\mu$, the last term is constant in $x$. Thus its minimization is the quadratic penalty method applied to the shifted constraint $c(x)=\mu u$.

Its practical advantage is that multiplier updates, typically $u^{+}=u-c(x)/\mu$, can drive the original constraint residual to zero while $\mu$ stays moderately sized. The plain quadratic penalty method generally needs $\mu$ very small to produce the same feasibility accuracy, exposing the conditioning problem in part (ii).