Let's say that I know that:

\sum_{i,j} a_{ij} \leq r, where 0 \leq a_{ij} \leq 0.5 and r \geq 0 and 1 \leq i,j \leq n

and I also know that

\sum_{i,j} y_{ij} \leq d, where y_{ij} \geq 0 and d \geq 0 and y is zero-diagonal and 1 \leq i,j \leq n

What do we know about?

\sum_{i,j} a_{ij} y_{ij} \leq ?

So far the best I can prove is

\sum_{i,j} a_{ij} y_{ij} \leq \frac{d}{2}

Is there a tighter bound?