An SQP-Filter Algorithm with an Improved Stochastic Gradient for Robust Life-Cycle Optimization Problems with Nonlinear Constraints

Abstract Solving a large-scale optimization problem with nonlinear state constraints has proven to be challenging when adjoint gradients are not available for computing the derivatives needed in the basic optimization algorithm employed. Here, we present a methodology for the solution of an optimization problem with nonlinear and linear constraints where the true gradients that cannot be computed analytically are approximated by ensemble-based stochastic gradients based on an improved stochastic simplex approximate gradient (StoSAG). For the most part, our discussion is focused on the application of our procedure to waterflooding optimization where the optimization variables are the well controls and the cost function is the life-cycle net present value (NPV) of production. The optimization algorithm used for solving the constrained optimization problem is sequential quadratic programming (SQP) with constraints enforced using the filter method. We introduce modifications to StoSAG that improve its fidelity, i.e., the improvements give a more accurate approximation to the true gradient (assumed here to equal the gradient computed with the adjoint method) than the approximation obtained using the original StoSAG algorithm. The improvements to the basic StoSAG vastly improve the performance of the optimization algorithm; in fact, we show that if the basic StoSAG is applied without the improvements, then SQP may yield a highly suboptimal result for optimization problems than many nonlinear state constraints involve.