In the context of discrete DC programming, Maehara and Murota (Mathematical Programming, Series A, 2014) posed the problem of determining the complexity of minimizing the difference of two M-convex set functions. In this paper, we show the NP-hardness of this minimization problem by proving a stronger result: maximizing the difference of two matroid rank functions is NP-hard. Maehara and Murota...

The conic structure of the convex cone of non-negative operator convex functions on (0,∞) (also on (−1, 1)) is clarified. We completely determine the extreme rays, the closed faces, and the simplicial closed faces of this convex cone.