minimization of log function

Question

I am trying to minimize a log function of 3 variables

$f(x,y,z) = a\log(x)+b\log(y)+c\log(z)$

such that $x+y+z=k$ for some constant $k$.

for $a,b,c > 0$ and $x,y,z > 1$

Now $f(x,y,z)$ is concave so Lagrangian method won't work to minimize. Any pointers on how to solve it? Also are there any standard programming libraries which can solve this?

Thanks!

Can you say anything about $a, b, c$, like they are all greater than zero? — Amzoti, Oct 26 '13 at 00:24
The function does not have a minimum, and its infimum is $-\infty$. Just set two of the variables equal to "nearly zero" and think what happens. — Alecos Papadopoulos, Oct 26 '13 at 00:54
Yes, that is true. Sorry, I am trying to solve for $x,y,z>1$ — aniket, Oct 26 '13 at 03:22

score 0 · Accepted Answer · answered Oct 26 '13 at 05:59

We will assume k > 3. f(x,y,z) = alogx + blogy + clogz = $log(x^ay^bz^c)$ Thinking geometrically, $(x^ay^bz^c)$ is the volume of a box with sides $x^a,y^b,z^c$. We know that this volume is maximized by take the 3 sides equal. Its minimum would be zero if we could take one side to be 0, but since x,y,z >1 we cannot do that. The best we can do is to take two sides as small as possible.

One of a, b, c, is smallest. Without loss of generality, suppose that is a. Then we can take y = z = 1 and x = k - 2, making $y^b$ and $z^c$ sides as small as possible, specificially 1.

Then f(x,y,z) = alog(k-2) and this should be your minimum.

minimization of log function

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