We consider the setting of publishing data without leaking sensitive information. We do so in the framework of Robust Local Differential Privacy (RLDP). This ensures privacy for all distributions of the data in an uncertainty set. We formulate the problem of finding the optimal data release protocol as a robust optimization problem. By deriving closed-form expressions for the duals of the constraints involved we obtain a convex optimization problem. We compare the performance of four possible optimization problems depending on whether or not we require robustness in i) utility and ii) privacy.