Knowing that a feasible solution exists and has a finite optimal solution

Question

I have the following linear programming problem:

constraints:

$x_1,x_2,x_3\geq200$

$0.45x_1+0.41x_2+0.5x_3 \leq 960$

$x_1+x_2+x_3 \leq 2000$

$ x_2+x_3 \leq x_1$

objective functions:

max $0.35 x_{1}+0.41 x_{2} + 0.37 x_3$

min $0.45x1+0.41x_2+0.5x_3$

How can I tell without solving the problem numerically that there is a feasible solution for both objective functions and a finite optimal solution?

Any advice about the theorem or the intuition would be greatly helpful!

Answer 1

As pointed out in the comments: We can easily construct a feasible solution, e.g. $x_1=400, x_2=x_3=200$ is feasible. It means that the feasible set is nonempty. The feasible set is bounded since $200 <= x_i <= 2000 $ holds for all variables. I.e. we want to maximize a linear function over a nonempty bounded set. From this it follows that the problem has a finite optimal solution.