I have the following optimization problem: \begin{equation} \begin{aligned} \max_{\vec{\alpha}} \quad & \sum_{i=1}^{M}\alpha_i f_i(\alpha_i)\\ \textrm{s.t.} \quad & \sum_{i=1}^{M} \alpha_i = \alpha \\ &\alpha_i \geq 0, i = 1,2,\ldots,M \\ \end{aligned} \end{equation} where $f_i(\alpha_i)$ is a decreasing function with respect to $\alpha_i$ , all $f_i(\alpha_i)$ belong to the same class function, e.g., the exponential function. And they may have different hyperparameters. $M$ is interger, $\alpha$ is a predefined value.
My question is: $f_i(\alpha_i)$ satisfy what conditions, that this problem has optimal solution,and what is the optimal solution?
Thank you for your help!
