I need to prove the following result:
There exists a unique solution to the system of equations
$$\alpha = \frac{I^T(\gamma e^{I\beta})}{\sum_{i=1}^{n}\gamma_ie^{(I\beta)_i}}$$ if and only if $\alpha = I^T \lambda$ ($\lambda \in (\mathbb{R}_+^*)^n$)
with $\alpha \in ]0,1[^p$ (a vector), $\gamma^T \in (\mathbb{R}_+^*)^n$ and $I$ a binary matrix (filled with 0 and 1 only) in $\mathcal{M}_{n\times p}$({0,1}). Of course $I^T$ denotes the transpose matrix if $I$. And we are in the case $n>p$. $I$ is a matrix of full rank.
$I^T(\gamma e^{I\beta})$ is a matrix-vector multiplication with $\gamma e^{I\beta}$ having components $\gamma_i e^{(I\beta)_i}$.
I want to use the convex conjugate of the log-sum-exp function to solve the problem but I am not comfortable enough in this field to write the right proof! Please help !!!
Here is an example using this approach
exemple using the convex conjugate with log-sum-exp
It's possible that this result comes from a theorem about composite convex conjugate. See A note on the Legendre-Fenchel transform of convex composite functions
