  None of the BROJA information decomposition measures $\mbox{SI}, \mbox{CI},
\mbox{UIy}, \mbox{UIz}$ are convex or concave over the probability simplex. In
this paper, we provide formulas for the sub-gradient and super-gradients of any
of the information decomposition measures. Then we apply these results to
obtain an optimum of some of these information decomposition measures when
optimized over a constrained set of probability distributions.
