A multiplier vector substitution technique based on pseudo-derivatives

An exact penalty technique based on an augmented Lagrangian function is used for solving equality constrained nonlinear optimization problems. For the Lagrange multiplier vector a substitution philosophy is used. To compute the stationary points of the augmented Lagrangian function, different versions are implemented. The first used the exact Jacobian matrix of the constraints. The second replaces the analytic Jacobian with a finite-difference approximation and the third defines a suitable pseudo-derivative approximation to the Jacobian. Numerical results show that two of them have similar performance. One can prove convergence to a stationary point of the augmented Lagrangian function regardless of the Jacobian approximation. We also prove that the pseudo-derivative problem formulation is equivalent to the penalty technique in sequential quadratic programming.