There is an exponential function on the denominator of my objective function
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My objective function has an exponential function on its denominator, so the optimizer cannot run. I tried to replace the denominator with a polynomial, but it seems that only quadratic polynomials can be used. For third and higher orders, the optimizer cannot run. My optimization problem is as follows
What should I do？ I request everyone's help. Thank you.ありがとうございます。
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Hi,
You can use auxiliary decision variables as below:
- \(y_j = -(\sum_{i=1}^{I} x_{ij} + \sum_{k=1}^{K} x_{kj}), ~~ \forall j \)
- \(z_j = e^{y_j}, ~~ \forall j\)
- \(u_j = \frac{2}{1+z_j} \rightarrow u_j + u_j z_j = 2, ~~\forall j\)
- \(v = \sum_{j=1}^{J} u_j - 1\)
- \(w = \frac{\sum_{j=1}^{J} (\sum_{i=1}^{I} x_{ij} r_i + \sum_{k=1}^{K} x_{kj} r_i )}{B_c}\)
- \(q = \frac{w}{v} \rightarrow w = qv\)
The objective function can then be just \(\max q\) and the above relationships should be implemented as constraints.
Best regards,
Maliheh
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