In the context of static real-time optimization (RTO) of uncertain plants, the standard modifier-adaptation scheme consists in adding first-order correction terms to the cost and constraint functions of a model based optimization problem. If the algorithm converges, the limit is guaranteed to be a KILT point of the plant. This paper presents a general RTO formulation, wherein the cost and constraint functions belong to a certain class of convex upper-bounding functions. It is demonstrated that this RTO formulation enforces feasible-side global convergence to a KKT point of the plant. Based on this result, a novel modifier adaptation scheme with guaranteed feasible-side global convergence is proposed. In addition to the first-order correction terms, quadratic terms are added in order to convexity and upper bound the cost and constraint functions. The applicability of the approach is demonstrated on a constrained variant of the Williams-Otto reactor for which standard modifier adaptation fails to converge in the presence of plant-model mismatch. (C) 2017 Elsevier Ltd. All rights reserved.