# Optimization problem with a bilinear constraint

I have an optimization problem with bilinear terms in the objective function and constraints, formulated as below:

$$\min C_1 x_1y +C_2 x_2y + C_3 x_3y$$

subject to:

$$z_1 + (A-1) x_1 y + A x_2 y + ...+ A x_Ny \leq B$$

$$z_2 + A x_1 y + (A-1) x_2 y + ...+ A x_Ny \leq B$$

$$z_m + A x_1 y + A x_2 y + ...+ (A-1) x_Ny \leq B$$

$$x_2 + \alpha < D$$

$$y + \beta < F$$

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.

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?

Many thanks

It’s just linear with $$u_i=x_iy \ \forall i$$.
• Apologies I should have mentioned that $x_i$ and $y$ are separately involved in other linear constraints in the formulation. I have edited the problem statement above. Oct 21 '20 at 0:14