# Interesting maxmin mixed integer/real quadratic optimization problem

I have the following problem:

$$\DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \argmax_{\underset{\lambda_k\in \mathbb{R}}{\sigma_q^2(k)\in \mathbb{R}}} \left[\min_{\underset{q\in \left[q_A,q_B\right]}{n_k\in \mathbb{Z}}} \sum_{k=1}^N \frac{1}{\sigma_q^2(k)}\left(q-n_k\lambda_k\right)^2 \right] \qquad \mbox{s.t.}\qquad \sum_{k=1}^N \frac{1}{\sigma_q(k)}=S$$

where the function to be minimized is quadratic both in the continuous variable $$q$$ and in the integer variables $$n_k$$ however I am having some problems translating it into a standard form. I tried to use the properties of the Hadamard product but that did not help much. Any ideas?

• Are there any constraints? If not, why not take $\sigma^2_q(k)$ to be (arbitrarily close to) zero, driving overall max objective to $\infty$, i.e., unbounded, i.e., not a very interesting problem. Sep 14 '19 at 0:22
• Good catch, there was indeed a missing constraint that I just added. Sep 14 '19 at 0:39
• @Arrigo Are the $\sigma_q(k)$ just notation or is $\sigma_q(k)$ some kind of function evaluated at the $k$'s? Is there any relationship among the different $\lambda_k$'s or are they independent? Jun 9 '20 at 4:56