Suppose I have a bunch of nonnegative integers $(a_{ijkl})_{1 \leq i \leq j \leq k \leq l \leq 17}$ such that *for all* 17-tuples nonnegative integers $w_t$ (for $1 \leq t \leq 17$) we have that $$\min_{1\leq i\leq j\leq k \leq l\leq 17}(a_{ijkl} + w_i + w_j + w_k+w_l) \leq \frac{4}{17}\sum_{1 \leq t \leq 17} w_t.$$ What can we say about the $a_{ijkl}$? Are they bounded? What are the optimal bounds? I would like to get as much information on the $a_{ijkl}$ as possible. This problem comes from a question in number theory I'm trying to answer, but as you can see, this has a smell of linear programming, of which I know nothing at all. Also, maybe some things can be done with a computer, but I'm not good with computers...

(EDIT) If you suppose moreover that for all $t$, at least one of the inequalities $a_{ijkl} \geq n$, where $n$ is the number of indices from $(i,j,k,l)$ which are equal to $t$ ($n \geq 1$), is *false*, can we get something more?

(MOTIVATION) This comes from a semistability condition in geometric invariant theory.