I have an optimization problem with bilinear terms in the objective function and constraints, formulated as below:
$ \min C_1 x_1y +C_2 x_2y + C_3 x_3y$
$z_1 + (A-1) x_1 y + A x_2 y + ...+ A x_Ny \leq B $
$z_2 + A x_1 y + (A-1) x_2 y + ...+ A x_Ny \leq B $
$z_m + A x_1 y + A x_2 y + ...+ (A-1) x_Ny \leq B $
$x_2 + \alpha < D $
$y + \beta < F $
where $A = 1/N$ and $z$, $x_n$, $z_m$, $\alpha$, $\beta$ and $y$ are unbounded decision variables. $A$ is a positive parameter between $0<A<1$.
Is there any way to reformulate the above into a SDP or other convex problem, or am I restricted to a non-linear optimization in this case?