Consider multi-valued function $F=(F_1,\cdots,F_d)$ where each $F_j$ is a multi-variate function mapping from $R^d$ to $R$. So $F$ is a function mapping from $R^d$ to $R^d$. Suppose each $F_j$ has simple forms such that their derivatives can be easily calculated
I'm wondering whether there are simple rules to verify that $F$ is convex, meaning that $F$ maps convex sets to convex sets. When $F$ is single valued then one can verify the positive semi-definiteness of its Hessian but for multi-valued $F$ I don't know whether there's a simple rule. for example,
- If $F(x)=\exp(Ax+b)$ where $\exp$ means element-wise exponentials. Is $F([0,1]^2)$ convex?
- If $F(x,y)=(e^{-ax}/(1+e^{-ax}+e^{-by}), e^{-by}/(1+e^{-ax}+e^{-by})$ for some $a,b>0$, is $F([0,1]^2)$ convex?