An Augmented Lagrangian Method With Flexible Penalty For Constrained Optimization

The augmented Lagrangian function is an extension of the quadratic penalty function or a combination of the Lagrangian function and the measurement of quadratic constraint violation by virtue of a penalty factor. The augmented Lagrangian function is usually used as a merit function whose purpose is to justify whether a current trial step can be accepted or not. The common feature of those methods is adopted one object, i.e., the augmented Lagrangian function. The sequence of the penalty factor generated is increasing monotonely on the concrete implementation of the algorithm.The bigger penalty factor may result in overflow in computation.In this paper, we propose a new algorithm falling in between penalty-type method and penalty-free method. In each iteration of the algorithm, the trial step is computed by minimizing a quadratic approximated model to the augmented Lagrangian function in the trust region. The model is just right a standard subproblem of trust region algorithms for unconstrained optimization which can be solved efficiently by many mature methods. We do not need to worry about the possible inconsistency between the linearized constraints and the trust region constraint, nor about linearly dependency of the gradients of the constraints. The solution of the subproblem is related to the penalty factor which is dependent on the messsage of the current iterate point. The sequence of the penalty factor is nonmonotone. The acceptable criteria of the new algorithm does not adopt augmented Lagrangian function as a merit function but the object function and the measurement of constraint violation to decide whether the trial step should be accepted or not.Under the weaker assumptions, we analyzed the well definedness of the algorithm and proved that there exists an accumulation point of the iterate sequence generated by the algorithm which is an infeasible stationary point, or linear independent constraint qualification does not hold at that accumulation point or is a first-order stationary point of the original problem. Finally, a preliminary numerical results are reported for some difficult optimization problems with nonlinear equality constraints.