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?

  • $\begingroup$ 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. $\endgroup$ Sep 14 '19 at 0:22
  • $\begingroup$ Good catch, there was indeed a missing constraint that I just added. $\endgroup$
    – Arrigo
    Sep 14 '19 at 0:39
  • $\begingroup$ @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? $\endgroup$
    – dhasson
    Jun 9 '20 at 4:56

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