Dealing with a non-convex problem

Question

I have the following objective function. The variables: $h_p$, $e_{trs}\left(h_p\right), w_{trs}\left(h_p\right)$ are all non-negative continuous. $T,R,S,\pi_{trs}$ are polynomially-sized sets. All other notations are non-negative parameters.

$$\max\sum_{\substack{t\in T,r\in R,\\s\in S}}\left[e_{trs}\left(h_p\right)+bw_{trs}\left(h_p\right)\right],$$

where

$$ \tag{1} e_{trs}\left(h_p\right)=C_{trs}\left(\sum_{p\in\pi_{trs}}l_p^{trs}\frac{30}{h_p}\right)^{\beta_{trs}} $$

and

$$\tag{2} w_{trs}\left(h_p\right)=\frac{C_{trs}^\prime}{2\left(1-\beta_{trs}\right)}\left[H-\left(\sum_{p\in\pi_{trs}}\frac{l_p^{trs}}{h_p}\right)^{\beta_{trs}-1}\right].$$

I would like to solve the problem by using cutting-plane method assuming (I haven't completely checked yet) both functions are convex in $h_p$ in some feasibility domain. So, I want to represent the functions $e_{trs}\left(h_p\right)=a+bh_p$ and $e_{trs}\left(h_p\right)=c+dh_p$, where $a,b,c,d$ are constants that are calculated through derivatives. I think, the barrier here is the max of the objective function. If it was min, I could use this approach (correct me if I am wrong) and add cuts iteratively. I thought instead of maximizing this objective, can I minimize $h_p$ and introduce the current objective function as a constraint? See below for a solution attempt. Would these two problems be equivalent?

$$\min\sum_{p\in\pi_{trs}}h_p$$ subject to (1) and (2). I feel like this is not correct. But, at the same time, I think, minimizing $h_p$ maximizes the value of the associated functions.

Answer 1

You want the two functions to be concave in $h_p$, since you are maximizing (convex would be correct if you were minimizing). As to whether minimizing the sum of the $h_p$ would be equivalent, it would not (in general ... I suppose that the two problems could accidentally have the same solution for a specific set of parameters). Any immediate problem with switching to the simpler objective function is that $\sum_p h_p$ remains constant if you increase one term by some amount and decrease another term by the same amount, whereas it is unlikely that change would be a wash in the original objective function.

Also, it's not clear to me that reducing $h_p$ increases the value of the $w$ function.