I would like to solve the following optimization problem:
\begin{array}{ll} \underset{x_{i}\geq0,\, \pi_{i}\in\{0,1\}}{\text{minimize}} & \displaystyle\sum_{i=1}^n x_i\\ \text{subject to} & \displaystyle\sum_{i=1}^n \pi_{i}\Big(\frac{q}{d+d \exp(ab-as_ix_i)}-c\Big) \geq t\\ &\sum x_i \leq l, . \end{array}
where $q,d,a,b,c,t,\text{and} ~l$ are real positive scalars and $s_1>s_2>…>s_n>0$.
The problem is non-convex due to the coupled binary variable and the sigmoid function in the first constraint. Also, if the binary variables $\pi_i$ are removed the problem still non-convex due the sum-of-ratio in the first constraint. How to solve this problem with\without the binary variables.