The combined method for solve the constraint optimization problems uses the augmented Lagrangian and line search technique

This paper presents an effective hybrid optimization methodology that combines the Augmented Lagrangian (AL) framework with the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm to address large-scale constrained problems. The method leverages the AL framework to impose quadratic and linear penalties on the constraints, transforming the constrained problem into a sequence of unconstrained subproblems, driven by the computational asymmetry of decomposition. Then, the L-BFGS method is used to make these smaller problems easier to solve. L-BFGS efficiently approximates the search direction with just operations per iteration by using a two-loop recursion with a limited history of gradient pairs. This avoids the problem of calculating the Hessian matrix. It is therefore very important for large-scale situations. The AL-BFGS design that comes out of this speeds up convergence in the inner minimization loop. The outer loop, on the other hand, keeps updating the Lagrange multipliers and changing the penalty parameter based on the primal and dual residuals. by making an efficient iterative solution and handling the inequality constraints correctly.