Maximize the following LP in a standard LP and minimize it in a canonical form.
$$\text{min}\;\;\; z=-2x_1+x_2+5x_3$$
$$\text{s.t.}\;\;\; x_1+x_2-3x_3 \le 6$$
$$2x_1+3x_2-5x_3=7$$
$$-2x_1+x_2+5x_3 \ge -3$$
$$x_1 \le 0,x_3 \ge 0$$
I think the linear programming can be maximized in a standard LP like this:
$$-\text{max}\;\;\; z=-2y-x_2-5x_3$$
$$\text{s.t.}\;\;\; -y+x_2-3x_3 +x_4= 6$$
$$-2y+3x_2-5x_3=7$$
$$2y+x_2+5x_3 -x_5= -3$$
$$y,x_3,x_4,x_5 \ge 0$$
And its minimization in a canonical form is:
$$\text{min}\;\;\; z=2y+x_2+5x_3$$
$$\text{s.t.}\;\;\; y-x_2+3x_3 \ge -6$$
$$-2y+3x_2-5x_3 \ge 7$$
$$2y-3x_2+5x_3 \ge -7$$
$$2y+x_2+5x_3 \ge -3$$
$$y,x_3 \ge 0$$
I would like to know how much of my work is correct.
Best Answer
Almost there.
According to the convention presented in Michel Goeman's note, we require all the variables to be nonnegative, hence we have to do a transformation to handle $x_2=x_2^+-x_2^-$ where $x_2^+, x_2^- \ge 0$.