# How can I deal with optimization problems that have a sum of functions of Z as a constraint when Z is the quantity to be minimized?

I have a problem where I have to minimize a certain quantity $$Z$$ subject to the following constraints:-

1. $$w_1 + w_2 + w_3 = 1$$
2. $$\frac{f_1(w_1*Z) + f_2(w_2 * Z) + f_3(w_3 * Z)}{Z} >= k$$

where $$k$$ is a known constant. $$f_1$$, $$f_2$$ and $$f_3$$ are non-linear functions that we have empirical curves for. (These curves are approximately logarithmic if that helps with the solution). $$w_1$$, $$w_2$$ and $$w_3$$ are weights whose optimal values are to be arrived at. I am aware of basic linear programming techniques. However, I was unable to reduce constraint #2 into a linear constraint.

Any help is welcome. Please let me know if any further details are required.

PS:- Solutions that use Python would be ideal. However, I'm more interested in the approach rather than language/package used.

1. $$Z$$ does not depend on the values of $$w_1, w_2, w_3$$
• Is it a typo in Eq.2, should be $f_3(w_3 * Z)$? – Maxim Umansky Nov 9 '20 at 5:07
• $Z$ depends on $w_1,w_2,w_3$ I suppose ? If the problem is $\min_{w_1,w_2,w_3} Z(w_1,w_2,w_3)$ subject to constraints (1) and (2), can you add this to your question ? – cfdlab Nov 9 '20 at 5:33
• @MaximUmansky You are correct. I have fixed the typo. Thanks! – Green Noob Nov 9 '20 at 6:40
• @cfdlab $Z$ does not depend on $w_i$. Added the same to my question. – Green Noob Nov 9 '20 at 6:53
• – SecretAgentMan Nov 9 '20 at 19:30

\begin{align}\min \>& Z \\ & \sum_i w_i = 1 \\ & \sum_i f_i(w_i\cdot Z)\ge k\cdot Z \\ & w_i \in [0,1] \end{align}
Getting rid of a division is always a good idea. If we can assume $$Z\gt0$$, then a slightly different formulation can look like:
\begin{align}\min \>& Z \\ & \sum_i w'_i = Z \\ & \sum_i f_i(w'_i)\ge k\cdot Z \\ & w'_i \ge 0 \end{align}
You can recover $$w_i$$ by $$w_i := w'_i/Z$$ (using the optimal values for $$w'_i$$ and $$Z$$).