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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!

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  • $\begingroup$ This may be related: cs.stackexchange.com/questions/13370/… $\endgroup$ – NoChance May 23 at 15:49
  • $\begingroup$ In general, detecting feasibility is just as hard as deciding if an LP is optimal, because adding the constraint $c^\top x \leq z_0$ to the system, and performing a binary search procedure to identify the largest $z_0$ such that the new system is feasible is equivalent to solving the original problem (i.e. there is a polynomial time reduction from optimization to feasibility checking). Consequently, there is typically no easy way of detecting feasibility unless solving the problem is easy. One exception: you can usually check whether "obvious" solutions (e.g. a vector of all 0s) are feasible. $\endgroup$ – rcorywright May 23 at 16:08
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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.

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