My problem is to optimize the following equation $$ \underset{\alpha,\beta,\boldsymbol{x}}{\text{minimize}} \text{ } F = \sum_{i=1}^{\beta}(x_i)^{\alpha} \\ \text{s.t. } F \geq e, \text{ } \sum_{i=1}^{\beta} x_i = d, \text{ } \alpha \in \mathbb{N}_{\leq n}, \text{ } \beta \in \mathbb{N}_{\leq d}, \text{ } \{x_i\}_{i=1}^{\beta} \in \mathbb{N}_{\leq d} $$ where the size of $\boldsymbol{x}$ depends on $\beta$, and all the variables are integers (>0). $\{e, d, n\}$ are constants, while the unknown variables include $\{\alpha,\beta,\boldsymbol{x}\}$.
I am not sure whether this optimization belongs to an integer nonlinear programming problem. And unfortunately, I am not familiar with such problem. Can anybody tell me what exactly this problem is and the complexity to solve it. In addition, is there any use-friendly libraries to solve this problem? Thanks!