For good performance in practice, real-time optimization schemes need to be able to deal with the inevitable plant-model mismatch problem. Unlike the two-step schemes combining parameter estimation and optimization, the modifier-adaptation approach does not require the model parameters to be estimated on-line. Instead, it uses information regarding the constraints and selected gradients to improve the plant operation. The dual modifier-adaptation approach presented in this paper drives the process towards optimality, while paying attention to the accuracy of the estimated gradients. The gradients are estimated from successive operating points generated by the optimization algorithm. The novelty lies in the development of an upper bound on the norm of the gradient errors, which is used as a constraint when determining the next operating point. The proposed approach is demonstrated via numerical simulation for both an unconstrained and a constrained problem.