I have to solve the following optimization problem using Groebner bases. The objective function is
$$x_1+x_2+x_3+x_4$$
with constraints
$$x_1+x_2 \ge 1, \qquad x_1+x_2+x_3 \ge 1, \qquad x_2+x_3+x_4 \ge 1, \qquad x_1+x_3+x_4 \ge 1$$
I tried in Maple, Mathematica and Matlab without success. Could someone help me with this? I have no idea.