Some weird "system" of inequalities in nonnegative integers. 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.
 A: Take $a_{iiii}=0$ for all $i=1,\dots,17$ and let the other $a$'s be arbitrary large. Then the inequality is satisfied.
Indeed, let $w_1, \dots, w_{17}$ be arbitrary nonnegative integers. Without loss of generality assume that 
$$\min \{ w_1, \dots, w_{17} \} = w_1.$$ 
Then
$$\min_{1\leq i\leq j\leq k\leq l\leq 17} (a_{ijkl} + w_i + w_j + w_k + w_l) \leq a_{1111} + 4w_1 = 4w_1 \leq \frac{4}{17} \sum_{t=1}^{17} w_t$$
as required.
Hence, $a_{ijkl}$ are not bounded in general.
A: I see this as more of a set selection problem or a combinatorial design problem than a number theory problem, unless there is something else about the $a_{ijkl}$ that is not being mentioned.
By setting 4 of the w's to 1 and the rest to 0, I can convince myself that 5 of the a's are zero.
Conversely, choose the a's anyway you like, but let me set 5 of them to zero: $a_{1111}$, and four others which partition the index set.  Then for any tuple of w's, either one of the four other a's I picked selects a below average subtuple of w's to sum, or they are all above average and $w_1$ is less than 1/17th of the sum.  Your inequality will be satisfied by this selection of a's and any w's, and it will not tell me a thing about the a's you picked.
Gerhard "Ask Me About System Design" Paseman, 2011.09.06
