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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.

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  • $\begingroup$ What value does n have? If not too big, perhaps BARON minlp.com/baron or BMIBNB under YALMIP yalmip.github.io/solver/bmibnb will work? If you can settle for a local minimum, you can try KNITRO. artelys.com/en/optimization-tools/knitro . if you know a feasible solution, that may help the optimizers. All of the above can do with or without binary restriction. $\endgroup$ Commented Jan 26, 2019 at 2:47
  • $\begingroup$ Normally, n<10. $\endgroup$ Commented Jan 27, 2019 at 12:59
  • $\begingroup$ For n < 10, my advice is to try a global optimizer, such as listed in my previous comment. $\endgroup$ Commented Jan 27, 2019 at 14:15

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