Estimation of Optimal Well Controls Using the Augmented Lagrangian Function with Approximate Derivatives

When efficient adjoint code for computing the necessary gradients is available, the augmented Lagrangian algorithm provides an efficient and robust method for constrained optimization. Here, we develop an augmented Lagrangian algorithm for constrained optimization problems where adjoint code is not available, and the number of optimization variables is so large that the approximation of gradients with the finite-difference method is not computationally feasible. Our procedure applies a pre-conditioned steepest ascent algorithm to maximize an augmented Lagrangian function which directly incorporates all bound constraints as well as all inequality and equality constraints. The pre-conditioned gradient of the augmented Lagrangian is estimated directly using a simultaneous perturbation stochastic approximation (SPSA) with Gaussian perturbations where the preconditioning matrix is a covariance matrix selected to impose a degree of temporal smoothness on the optimization variables, which, for the specific application considered here, are the well controls. Our implementation of this augmented Lagrangian method is applied to estimate the well controls which maximize the net present value (NPV) of production for the remaining life of a given oil reservoir.