We derive and implement a new optimization algorithm on the basis of 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 in which 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. QIM-AG represents a significant modification of an optimization algorithm—new unconstrained optimization algorithm (NEWUOA)—derived fairly recently in the mathematical literature. Production-optimization examples show that QIM-AG results in a higher NPV in fewer iterations than is obtained with NEWUOA. QIM-AG is also compared with two other optimization algorithms that use an AG—namely, a slightly improved implementation of ensemble optimization presented in this paper and a new implementation of simultaneous perturbation stochastic approximation (SPSA).