Consider the problem:
$$maximize: Z = x_1+2x_2$$
subject to: $$x_1+x_2 \leq 8$$ $$-x_1+2x_2 \leq 2 $$ $$x_1-x_2 \leq 4$$
I know that this problem can be solved by using Branch-and-Bound method, but if we change the maximizing function to: $$ Z=x_1^2+ 2x_2$$ could we use again the Branch-and-Bound method to solve this problem?
I think (but I am not sure) that we are allowed to use Branch-and-Bound method since the constrains are still linear. And if we cannot use Branch-and-Bound method, then which method would work to solve this problem?
Thank you in advance for any advice
I'm going to assume that $x_1$ and $x_2$ are integer variables. The question does not specify this, but discussion of branch-and-bound implies it.
What most people think when they hear "branch and bound" or "branch and cut" is binary tree search with objective bounds from LP relaxations. That would not apply here, because the objective function is not linear. There are extensions that allow you to minimize a convex quadratic objective or maximize a concave quadratic objective, but you are trying to maximize a convex quadratic, so those extensions would not apply.
Because the problem is so small, and the feasible region is bounded, you could just enumerate all feasible values of $x_1$, solve a tiny LP for $x_2$ (by inspection), and pick the winning answer. More generally (i.e., for higher dimensional problems), you would need to look into nonconvex quadratic program solvers.