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We have to minimize $f(x,y,z) = a x + b ln(y) + c ln(z)$ with linear constraints of the form $\ d < x,y,z < e $, where $\ a, b, c,d $ and $\ e$ are constants.

$\ f(x,y,z) $ is a strictly concave function because its Hessian matrix turns out to be negative definite, assuming that $\ a, b, c,d $ and $\ e$ are positive.

Is there a way to convert it into a convex function that would allow a global minima? If not, which solvers can be used to optimize a concave function?

user323707
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  • Are you sure that the domain should be open? Anyway $f$ is separated and each summand is strictly increasing. So you can find the infimum by looking. On the open domain, it won't attain its infimum. – user251257 Mar 28 '16 at 20:47

1 Answers1

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If your constants are positive, then, f is increasing in $x$, $y$ and $z$. Therefore, minimizing $x$, $y$ and $z$ minimizes $f$.

If the constraints are not strict (ie : $d \le x,y,z\le e$), $f$ admits a minimum on this domain which is $f(d, d, d)$

Joseph Budin
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