Let
$x_1 + x_2 + ... + x_n=y_1$
and
$a_1x_2 + a_2x_2 + ... + a_nx_n = y_2$ the two addtional constraints.
The provisional objective function would be $|y_1-T|+|y_2-At\cdot t|$ to calculate the sum of the two differences. It doesn´t play a role wether the differences are positive or negative or not.
To get rid off the absolute values, you have to define:
- $y_3=y_1-T$ and $y_4=y_2-At\cdot t$
The objective function is now $|y_3|+|y_4|$
- $|y_3|=y_3^++y_3^-$ and $|y_4|=y_4^++y_4^-$
And finally the objective function is
$\texttt{min} \ \ y_3^++y_3^-+y_4^++y_4^-$
Transforming $y_3$ and $y_4$
$y_3=y_3^+-y_3^-$ and $y_4=y_4^+-y_4^-$
The two additional constraints are
$x_1 + x_2 + ... + x_n=y_3^+-y_3^-+T$
$a_1x_2 + a_2x_2 + ... + a_nx_n = y_4^+-y_4^-+At\cdot t$
The variables are $x_i, y_3^+,y_3^-, y_4^+,y_4^- \geq 0$
Best Answer
it's non linear. it looks like separation of variables via substitution may work, introducing additional constraints, see p15 of:
[1]http://web.mit.edu/15.053/www/AMP-Chapter-13.pdf