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?


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 $y$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.