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Let us be given a function $f(x_1,\dots,x_{10})$ of multiple variables $x_1,\dots,x_{10}$ given that $\sum_{i=1}^{10} x_i \leq 7$. How do we solve the following problem?

$$ \begin{equation} \begin{aligned} \max_{\{x_i\}, i=1,\dots,10} \quad & f(x_1,\dots,x_{10})\\ \textrm{s.t.} \quad & \sum_{i=1}^{10} x_i \leq 7\\ & x_i \in X_i, X_i = [0,10] \\ \end{aligned} \end{equation} $$

I with my professor are trying to work on building a neural network in such a way that we have multiple output from each layer. For example there are $n$ neurons in the $i^{th}$ layer, there will be, let us say $k$ outputs of each size $n$. Similarly, the next layer will also have same number of output and we will look at all the possible ways to connect $k$ outputs of the $i^{th}$ layer and $k$ outputs of the $(i+1)^{st}$ layer. And we take the max of the output in each layer while updating the loss function. My professor told me that this boils down to finally become a DP problem.

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  • $\begingroup$ Any additional information, such as the contribution of each $x_i$ is independent of all $x_j, j\ne i$ or all $x_i$s are integral? $\endgroup$
    – greybeard
    May 10 at 18:42
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    $\begingroup$ If the variables are integers, there are only 19448 options to try. $\endgroup$ May 10 at 18:52
  • $\begingroup$ Please edit the question to indicate whether the variables must be integers, or if not, what domain they come from. $\endgroup$
    – D.W.
    May 13 at 6:32
  • $\begingroup$ "$k$ outputs of each size $n$": what ?? $\endgroup$ May 13 at 8:42
  • $\begingroup$ If all $X_i$ are identical, why $i$ ? And due to the sum and positiveness constraints, $x_i\le 7$ must hold anyway. $\endgroup$ May 13 at 8:43

2 Answers 2

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There are only 19448 combinations of $x$'s that meet the constraints. If $f$ is arbitrary, the best you can do is enumerate all 19448 combinations and see which leads to the largest value of $f$. There is no faster algorithm. Dynamic programming does not seem relevant.

If you know something about the structure of $f$ (e.g., it is a separable function), then it might be possible to do better.

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I do not see a dynamic programming solution. I highly doubt there is one (especially, a polynomial time).

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