Before revision:  Nevertheless, we show that when the buyer type is known, the problem is equivalent to finding the concave closure of a function  . Based on this observation, we prove some properties of the optimal mechanism, which allow us to solve  the optimal mechanism by a convex program ( with exponential size in general, polynomial size for special cases).
After revision:  Nevertheless, we show that when the buyer type is known, the problem is equivalent to finding the concave closure of a function . 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 ( with exponential size in general, polynomial size for special cases).