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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 $0<A<1$.

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?

Many thanks

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It’s just linear with $u_i=x_iy \ \forall i$.

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  • $\begingroup$ 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. $\endgroup$ Oct 21 '20 at 0:14
  • $\begingroup$ @Fenderman2014 Same answer! $\endgroup$
    – Khue
    Oct 21 '20 at 0:20

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