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Any ideas how to solve the following problem:

$$Minimize: |F(x,y)|+|G(x,y)|$$ s.t. $x<A, y<B$

where $$F(x,y)=ax^2+by^2+cx+dy+e$$ $$G(x,y)=fx^2+gy^2+hx+iy+j$$

and $A,B$ are known constants.

Any help would be appreciated.

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You can modify this program into a QCLP as follows,
$ Minimize : z + w $ such that $|F(x,y)| \leq z$, $|G(x,y)| \leq w$, $x < A$, $y < B$, $z,w \geq 0$
which basically becomes,
$ Minimize : z + w $ such that $ -z \leq F(x,y) \leq z$, $-w \leq G(x,y) \leq w$, $x < A$, $y < B$, $z,w \geq 0$

Now, you can solve this using any of the standard techniques.

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