I would like to formulate the following Optimization problem. My question is focused on the constraint. Given a "typical" objective function, e.g.:
$$ \min c^T v $$
s.t.
$$ 0 = a_1 v_1 - a_2 v_2 + a_3 v_3 + a_4 v_4 +\dotsb $$
or written alternatively:
$$ \begin{gather} 0 = a^T Q v \\ 0 \le b_L ≤ v \le b_U \\ 0 < d_L ≤ a ≤ d_U, \end{gather}$$
where $v$ and $a$ are variable vectors ($a$ is strictly positive and $v$ is positive); $b$ and $d$ are upper and lower bound vectors. I recognize that this constraint is bilinear. I'm hoping this problem is a recastable as LP or MILP given the equalities, and the positive variables. Is there an easy LP re-casting in this case?
I should mention that keeping the number of new constraints down to a minimum is at a premium. The McCormick convex relaxation seems like overkill (each constraint being recast with 4 new ones). Also, if possible I would like to implement this in IBM ILOG CPLEX.