I would like to solve a problem obtained from a LP by adding a second degree term to its utility.
A simple example would be the following (with $c_i > 0$): $$ \min xy - c_1 z_1 - c_2 z_2 \\ \textrm{s.t. } x = z_1 + c_3, \\ y = z_2 + c_4, \\ x, y \geq 0. $$
I believe the problem cannot be converted into a geometric program (due to negative coefficients) nor a mixed linear geometric program (as the variables are not partitioned).
I tried to obtain a convex problem by introducing new variables $X := \log x, Y := \log y$ for $$ \min t - c_1 z_1 - c_2 z_2 \\ \textrm{s.t. } t \geq e^{X+Y}, \\ e^X = z_1 + c_3, \\ e^Y = z_2 + c_4. $$ I thought about relaxing the equalities ($e^X \leq z_1 + c_3$, $e^Y \leq z_2 + c_4$) and compensate with a penalty term in the utility -- but I stuck here.
Is there an efficient way to solve such problem?
