# Sanity check about a linear programming problem

given the linear program:

minimize $$x+y$$

subject to,

$$ax+by \leq 1$$

$$x,y \geq 0$$

I need to find real numbers $$a,b \in \mathbb{R}$$ such that the program (a) is infeasible, (b) is unbounded, and (c) has a unique optimal solution.

Since we are asked to minimize the sum of two non-negative numbers the solution for all $$a,b \in \mathbb{R}$$ is $$(x,y)=(0,0)$$, right?

I cannot see any potential values of $$a,b$$ which would make the program infeasible or unbounded, because we are always looking to minimize the sum of non-negative numbers.

• Hint: if $x,y \ge 0$ then it cannot be that $x+y \le -1$. – Yuval Filmus Apr 4 '19 at 16:45
• @YuvalFilmus What values of $a,b$ transform the constraint into the one you have shown? If we take $a,b < 0$ then the constraint is equivalent to $ax+by \geq -1$, but even for that (0,0) is an optimal solution. If we take $a > 0$ and $b < 0$, then the constraint is equivalent to $-ax+by \geq -1$ and again we have that (0,0) is the optimal solution. – Matthew Riley Apr 4 '19 at 17:03
• Well then, perhaps there is a mistake in the question. Perhaps you can prove that the program is feasible and bounded for all $a,b$. – Yuval Filmus Apr 4 '19 at 17:04