We consider the real-time optimization of static plants and propose a generalized version of the modifier-adaptation strategy that relies on second-order adaptation of the cost and constraint functions. We show that second-order adaptation allows checking whether a (local) plant optimum is reached upon convergence. A sufficient convergence condition that is applicable to first- and second-order modifier-adaptation schemes is proposed. We also discuss how second-order updates can lead to SQP-like model-free RTO schemes. The approach is illustrated via the simulated example of a continuous reactor.