Consider the following function $f(x,y)$ :

$$f(x,y)=\frac{c_1}{c_2x+c_3y}+ \frac{c_4}{c_5-c_6x-c_7y} + c_8$$

Where $c_1,c_2,c_3,c_4,c_5,c_6,c_7,c_8>0$.

The optimization problem is:

minimize $f(x,y)$

s.t $x>0\\ y>0$

After solving $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$, I got



By replacing $x$,


Now, from the expression, both $x$ and $y$ can not be positive at the same time and these values make the denominator of the first term $0$. How to solve this problem?


There's no mimimum.

For $x,y > 0$, $f$ is unbounded below.

Just choose $\epsilon$ positive and sufficiently small, and take $x,y > 0$ on the line $c_6x+c_7y=c_5 + \epsilon$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.