I need to minimize the following function $$f(x,y)= \frac{a}{x}+\frac{bx}{y}+\frac{cy}{x}+dy+\frac{e}{y}$$ where $a,b,c,d$, and $e$ are positive constants, and $x$ and $y$ are both strictly positive.
I believe the best way is to prove that $f(x,y)$ pseudoconvex, or invex? (I could not do this). However, This is what I have been to:
From the Hessian matrix, the function is not convex for all possible values of $a,b,c,d$, and $e$.
Setting the gradient of $f(x,y)$ to $0$ gives a unique solution of x and y.
I also noticed that the function goes to infinity when $x=0$ or $y=0$ or $x\to\infty$ or $y\to\infty$.
Hence I conclude that the stationary point is global minimum
- Can you help me characterize the function? (pseudoconvex, invex ..?)
- What do you think of my approach? is it valid?
This is a geometric program. Let $x = e^u$, $y=e^v$, then the function is convex in $u$ and $v$.