I was going through the following paper titled "Adversarial Classification Under Differential Privacy".
https://www.ndss-symposium.org/wp-content/uploads/2020/02/23047-paper.pdf.
In this paper, the authors have solved the Lagrangian of an objective function in Theorem 3.1 (page number 4-5). They have not provided the complete proof as shown on page 5. I am trying to solve the Lagrangian of the objective function as stated in the paper. However, I am not able to get the result as expected. I am stuck at how "the derivative of the Lagrangian with respect to $\alpha$ and set $\alpha = 0$ (the optimal point), for all possible $p(r)$" is leading to the final result (page 5). It will be really helpful if someone can help me solve the Lagrangian to obtain the optimality condition.