Function Meaning Domain Sign Curvature Monotonicity
abs(x) $ |x| $ $ x \in \mathbf{R} $  positive  convex

 incr. for $ x \geq 0 $

 decr. for $ x \leq 0 $

entr(x) $ \begin{cases} -x \log (x) & x > 0 \\ 0 & x = 0 \end{cases} \\ $ $ x \geq 0 $  unknown  concave

None

exp(x) $ e^{x} $ $ x \in \mathbf{R} $  positive  convex

 incr.

geo_mean(x1,...,xk) $ (x_{1} \cdots x_{k})^{1/k} $ $ x_{i} \geq 0 $  positive  concave

 incr.

huber(x) $ \begin{cases} 2|x|-1 & |x| \ge 1 \\ |x|^{2} & |x| < 1 \end{cases} \\ $ $ x \in \mathbf{R} $  positive  convex

 incr. for $ x \geq 0 $

 decr. for $ x \leq 0 $

inv_pos(x) $ 1/x $ $ x > 0 $  positive  convex

 decr.

kl_div(x,y) $ x \log (x/y)-x+y $ $ x,y > 0 $  positive  convex

None

log(x) $ \log(x) $ $ x > 0 $  unknown  concave

 incr.

log_sum_exp(x1,...,xk) $ \log \left(e^{x_{1}} + \cdots + e^{x_{k}} \right) $ $ x \in \mathbf{R}^{k} $  unknown  convex

 incr.

max(x1,...,xk) $ \max \left\{ x_{1}, \ldots , x_{k} \right\} $ $ x \in \mathbf{R}^{k} $ max(sign(arguments))  convex

 incr.

min(x1,...,xk) $ \min \left\{ x_{1}, \ldots , x_{k} \right\} $ $ x \in \mathbf{R}^{k} $ min(sign(arguments))  concave

 incr.

norm1(x1,...,xk) $ |x_{1}| + \cdots + |x_{k}| $ $ x \in \mathbf{R}^{k} $  positive  convex

 incr. for $ x \geq 0 $

 decr. for $ x \leq 0 $

norm2(x1,...,xk) $ \sqrt{x_{1}^{2} + \cdots + x_{k}^{2}} $ $ x \in \mathbf{R}^{k} $  positive  convex

 incr. for $ x \geq 0 $

 decr. for $ x \leq 0 $

norm_inf(x1,...,xk) $ \max \left\{ |x_{1}|, \ldots, |x_{k}| \right\} $ $ x \in \mathbf{R}^{k} $  positive  convex

 incr. for $ x \geq 0 $

 decr. for $ x \leq 0 $

pos(x) $ \max \{x,0\} $ $ x \in \mathbf{R}$  positive  convex

 incr.

pow(x,p), $\text{ } p \geq 1 $ $ x^{p} $ $ x \geq 0 $  positive  convex

 incr.

pow(x,p), $\text{ } 0 < p < 1 $ $ x^{p} $ $ x \geq 0 $  positive  concave

 incr.

pow(x,p), $\text{ } p \leq 0 $ $ x^{p} $ $ x > 0 $  positive  convex

 decr.

quad_over_lin(x,y) $ x^{2}/y $ $x \in \mathbf{R}$, y > 0  positive  convex

 incr. for $ x \geq 0 $

 decr. for $ x \leq 0 $

 decr. in y

sqrt(x) $ \sqrt{x} $ $ x \geq 0 $  positive  concave

 incr.

square(x) $ x^{2} $ $ x \in \mathbf{R} $  positive  convex

 incr. for $ x \geq 0 $

 decr. for $ x \leq 0 $