Supermodular

$f:R^k \to R$

is supermodular if

$f(z \lor z') + f(z \land z') \geq f(z) + f(z')$

for all z, z' $\isin$ R^k, where z $\vee$ z' denotes the component-wise maximum and z $\wedge$ z' the component-wise minimum of $z$ and $z'$ .

If −f is supermodular then f is called submodular, and if the inequality is changed to an equality the function is modular.

If f is smooth, then supermodularity is equivalent to the condition

$\frac{\partial ^2 f}{\partial z_i \partial z_j} \geq 0 \mbox{ for all } i \neq j.$