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I need to find the vector $\hat{n}=[n_1 \; n_2 \; \cdots \; n_N]$, all integers,such that $1\leq n_i \leq Z$, $\forall i$ (with $Z$ also integer) that minimizes the following function:

$F=\frac{1}{N}\sum^N_{i=1}\left(1-\prod^N_{j=1,j\neq i}\left(1-\alpha_{ij}\frac{n_j}{Q}\right)\right)^{n_i}$,

where $Q>>Z$ is also an integer, and $0\leq \alpha_{ij}\leq 1$, $\forall i$ $\forall j$.

Any hint on promising approaches for solving this problem?

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  • $\begingroup$ It depends on why you are interested in solving this and/or what kind of "solution" is acceptable. Are you looking for an algorithmic way to obtain the solution? Do you need theoretical guarantees about this algorithm? Will you actually be implementing this algorithm, and thus need to consider things like numerical stability? $\endgroup$ Commented Apr 18, 2024 at 0:23
  • $\begingroup$ But the problem is pretty interesting. Since the base of each term is less than 1 regardless of how big the $n_j$'s got, I originally thought that that the solution would be to make each $n_i$ as large as possible. But playing around with it I have realized this is not the case. $\endgroup$ Commented Apr 18, 2024 at 0:23
  • $\begingroup$ Thanks. I am wondering whether it is possible at all to find analytically the values of the vector $\hat{n}$ that minimize $F$; otherwise a good approach for finding a good solution. $\endgroup$ Commented Apr 18, 2024 at 6:38
  • $\begingroup$ How large are $Z$ and $N$? Brute force enumeration would require testing $Z^N$ solutions. $\endgroup$ Commented Apr 18, 2024 at 16:10
  • $\begingroup$ $Z$ can be in the order of 10, while $N$ can even be 100... brute force is not an option $\endgroup$ Commented Apr 18, 2024 at 17:59

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