Submodular function

In mathematics, a function

$f:R^{k}\mapsto R$

is supermodular iff

$z,z'\in R^{k}\Longrightarrow f(z\lor z')+f(z\land z')\geq f(z)+f(z').$

Where $z\lor z'$ denotes the component-wise maximum and $z\land z'$ 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, supermodularity is equivalent to the condition

$\partial ^{2}f/\partial z_{i}\partial z_{j}\geq 0$ for all $i\neq j$.