In mathematics, a function
f
:
R
k
→
R
{\displaystyle f\colon \mathbb {R} ^{k}\to \mathbb {R} }
is supermodular if
f
(
x
↑
y
)
+
f
(
x
↓
y
)
≥
f
(
x
)
+
f
(
y
)
{\displaystyle f(x\uparrow y)+f(x\downarrow y)\geq f(x)+f(y)}
for all
x
{\displaystyle x}
,
y
∈
R
k
{\displaystyle y\in \mathbb {R} ^{k}}
, where
x
↑
y
{\displaystyle x\uparrow y}
denotes the componentwise maximum and
x
↓
y
{\displaystyle x\downarrow y}
the componentwise minimum of
x
{\displaystyle x}
and
y
{\displaystyle y}
.
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 twice continuously differentiable, then supermodularity is equivalent to the condition
∂
2
f
∂
z
i
∂
z
j
≥
0
for all
i
≠
j
.
{\displaystyle {\frac {\partial ^{2}f}{\partial z_{i}\,\partial z_{j}}}\geq 0{\mbox{ for all }}i\neq j.}
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