We derive and implement a new optimization algorithm based on a quadratic interpolation model (QIM) for the maximization (or minimization) of a cost or objective function. Although we have also applied the algorithm in other petroleum engineering applications, this paper restricts the algorithm's application to the production optimization step of closed-loop reservoir management where the objective function is the net present value (NPV) of production from a given reservoir. The new algorithm does not require a gradient calculation with an adjoint method but does use an approximate gradient (AG). Thus, the general optimization algorithm is referred to as QIM-AG. At each iteration, the QIM-AG algorithm builds a quadratic model which approximates the objective function using the approximate gradient and interpolation points. As the number of interpolation points is usually far less than the number of coefficients in the quadratic model, the extra degrees of freedom are used to minimize the Frobenius norm of the Hessian matrix of the quadratic model. The quadratic model is then maximized using a trust-region method. As more objective function evaluations become available during iteration, the quadratic model is updated with new interpolation points and an approximate gradient. The performance of the new algorithm is compared with the performance of several algorithms which are either derivative-free or use only a derivative approximation which can be calculated without using the adjoint method. The derivative-free optimization algorithms considered for comparison include NEWUOA (New Unconstrained Optimization Algorithm), SID-PSM (Pattern Search Method guided by Simplex Derivatives), PSO (Particle Swarm Optimization), EnOpt (ensemble-based optimization) and SPSA. A brief conceptual summary of these other derivative-free algorithms is given and it is shown that the approximate gradient implicitly used in EnOpt can be directly related to the simplex gradient used in SID-PSM.
For a set of three distinct examples, the QIM-AG algorithm generally outperforms the other derivative-free optimization algorithms. Specifically, QIM-AG obtains an estimate of optimal controls that gives the highest NPV in two of the three examples and within 0.1% of the highest value in the third example.