augmented Lagrangian

A penalty function combined with a multiplier estimate, letting the penalty weight stay fixed instead of growing without bound.

For an equality constraint $h(x)=0$ , the augmented Lagrangian is $L_A(x,\nu;\rho)=f(x)+\nu\,h(x)+\frac\rho2h(x)^2$ , combining the ordinary Lagrangian's multiplier term with a quadratic penalty on the same constraint. Alternating an $x$ -minimization with the multiplier update $\nu_{k+1}=\nu_k+\rho\,h(x_k)$ converges to the true multiplier at a fixed, modest $\rho$ , avoiding plain penalty's need for $\rho\to\infty$ .

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