I've come across a question which I was not able to solve I would appreciate if someone could help me out here.
Q) Given the constraints,
$$x \ge 0$$ $$y \ge 0$$ $$x + y \le1$$
which of the following is likely to be an optimal solution if the objective is to minimise $~ax + by~$ for any real numbers $~a~$ and $~b~$?
These are the options available: $$\text{a) }(0,0)$$ $$\text{b) }(1,1)$$ $$\text{c) }(1,0)$$ $$\text{d) }(0,1)$$ $$\text{e) }(a,b)$$
In a linear optimisation problem, the optimal solution (minimum or maximum) are always on the vertex of the constraint polygon.
In the problem here, the constraint polygon is a triangle whose vertex are at $(0,0)$, $(1,0)$ and $(0,1)$. On which vertex is the minimum depends on the value of $a$ and $b$.
Let first evaluate the function $ax+by$ at these points. $$a\cdot 0 + b \cdot 0 = 0$$ $$a\cdot 1 + b \cdot 0 = a$$ $$a\cdot 0 + b \cdot 1 = b$$
If $a,b > 0$, then the minumum will be at $(0,0)$
If either $a$ or $b$ is below $0$ then the minimum will be at smallest value. E.g. if $a<0$ and $a < b$ then the minimum will be at $(1,0)$
There are three special cases to consider.
1- If $a=0$ and $b>0$, then any point on the lattice between $(0,0)$ and $(1,0)$ is a minimum of the function.
2- If $b=0$ and $a>0$, then any point on the lattice between $(0,0)$ and $(0,1)$ is a minimum of the function.
1- If $a<0$ and $b<0$ and $a=b$, then any point on the lattice between $(1,0)$ and $(0,1)$ is a minimum of the function.