See also: Constrained Optimization Nonlinear Programming

*Augmented Lagrangian method* is one of the algorithms in a class of methods for constrained optimization that seeks a solution by replacing the original constrained problem by a sequence of *unconstrained* subproblems. Also known as *the method of multipliers*, the augmented Lagrangian method introduces explicit Lagrangian multiplier estimates at each step.

Augmented Lagrangian algorithms are based on successive minimization of the augmented Lagrangian \(\mathcal{L}_A\) with respect to \(x\), with updates of \(\lambda\) and possibly occurring between iterations. An augmented Lagrangian algorithm for the constrained optimization problem computes \(x_{k+1}\) as an approximate minimizer of the subproblem

\[\min \{ {\mathcal{L}_A(x, \lambda_k; \nu_k) : l \leq x \leq u} \},\]
where

\[\mathcal{L}_A(x, \lambda; \nu) = f(x) + \sum_{i \in \mathcal{E}} \lambda_i c_i(x) + \frac{1}{2} \sum_{i \in \mathcal{E}} \nu_i c_i^2(x)\]
includes only the equality constraints. Updating of the multipliers usually takes the form

\[\lambda_i \leftarrow \lambda_i + \nu_i c_i (x_k).\]

This approach is relatively easy to implement because the main computational operation at each iteration is minimization of the smooth function \(\mathcal{L}_A\) with respect to \(x\) subject only to bound constraints. A large-scale implementation of the augmented Lagrangian approach can be found in the LANCELOT package, which is available on the NEOS Server. LANCELOT solves the bound-constrained subproblem by using special data structures to exploit the (group partially separable) structure of the underlying problem. The OPTIMA and OPTPACK libraries also contain augmented Lagrangian codes.