Real-time optimization (RTO) via modifier adaptation is a class of methods for which measurements are used to iteratively adapt the model via input-affine additive terms. The modifier terms correspond to the deviations between the measured and predicted constraints on the one hand, and the measured and predicted cost and constraint gradients on the other. If the iterative scheme converges, these modifier terms guarantee that the converged point satisfies the Karush-Kuhn-Tucker (KKT) conditions for the plant. Furthermore, if upon convergence the plant model predicts the correct curvature of the cost function, convergence to a (local) plant optimum is guaranteed. The main advantage of modifier adaptation lies in the fact that these properties do not rely on specific assumptions regarding the nature of the uncertainty. In other words, in addition to rejecting the effect of parametric uncertainty like most RTO methods, modifier adaptation can also handle process disturbances and structural plant-model mismatch. This paper shows that the use of a convex model approximation in the modifier-adaptation framework implicitly enforces model adequacy. The approach is illustrated through both a simple numerical example and a simulated continuous stirred-tank reactor.