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I have a function which I suspect is jointly convex, but have a difficult time proving it, especially since the Hessian is messy.

The function is $f(y_i,i=1.2,\ldots,N)=\sum_i l_i w_i + y_i$, where $$w_i=a_i \exp(-y_i/a_i)\Pi_{j\in N_i} b_j w_j$$

$a_i$s, $b_j$s and $l_i$s are constants and $N_i$ is some subset of $\{1,2,\ldots,N\}$. Is it possible to argue the convexity of this function by some other way?

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A sum of convex functions is convex, so if the $l_i \ge 0$ it is sufficient for all $w_i$ to be convex functions of $y$.

If you require $w_i > 0$ (and all the constants $a_i$ and $b_j$ are positive), taking logarithms gives you a system of linear equations in $\log(w_i)$ and $y_i$, so (if there is a solution) you get $\log(w_i)$ as affine functions of $y_i$. Exponentiating, $w_i = c_i \exp(\sum_j d_{ij} y_j)$ for some constants $c_i$ and $d_{ij}$ with $c_i > 0$. This is a convex function of $y$.

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