Before revision:  In this case, we prove that finding the optimal advertising strategy is equivalent to finding the concave closure of a function, which is NP-hard in general . Based on this observation, we prove some properties of the optimal mechanism, which allow us to solve for the optimal mechanism by a convex program (of exponential size in general, polynomial size for special cases).
After revision:  In this case, we prove that finding the optimal advertising strategy is equivalent to finding the concave closure of a simple function. The function is a product of two quantities, the likelihood ratio and the cost of uncertainty . Based on this observation, we prove some properties of the optimal mechanism, which allow us to solve for the optimal mechanism by a convex program (of exponential size in general, polynomial size for special cases).