I've run into a wall in formulating a convex program for finding equilibria in an economic model I've been working on. I have one bilinear/quadratic constraint left that is getting in my way. Any suggestions/tricks/help would be much appreciated.
The problem boils down to finding a convex formulation of the following bilinear constraint:
Suppose you have vectors $p_1,p_2,x_1,x_2,y_1,$ and $y_2$. We know vectors $p_1, p_2$ are positive and that the rest are non-negative. Then the constraint is $$p_1 x_1 + p_2 x_2 \leq p_1 (y_1 + y_2)$$ Also, it is worth noting that we know that at equilibria (e.g. solutions to the program) we know that the inequality is in fact an equality.
In summary, is there some way of rewriting the above bilinear constraint as a convex constraint?