From the proof of Theorem 8 in Hashimoto et al. 2016, I hardly don't understand a small thing about the minima of the weighted energy function.
Consider the objective,
$$\min_{\lambda_{ij}} \;C_{ij} \log \left( \frac{\exp(-\lambda_{ij})}{\sum_k\exp(-\lambda_{ik})} \right)$$
for scalar values $C_{ij}$ and $\lambda_{ij}$.
Then the minima of the objective is given as $\lambda_{ij}^* = -\log(C_{ij}) + a_i$ with un-identifiable value $a_i$ by differentiation.
My question is how we can obtain the minima like that by differentiation. Thanks in advance for your answer.