I am dealing with an optimization problem in which I want to find $x$ that minimizes $y(x)$. Instead of resorting to numerical optimization, I would like to find a closed-form solution. So, after taking the derivative of $y(x)$, I arrived to the following equation, for which I am not able to find a closed-form solution:

$\left[\frac{\alpha}{x}-\beta\right]e^{-\kappa x}=\gamma$

where $\alpha$, $\beta$, $\kappa$ and $\gamma$ are positive real-valued constants. Also, the solution for $x$ should be real-valued and greater than $0$. Using mathematica, it seems that the equation can be solved when isolating each one of the two LHS terms, but not when both are considered together...

Any help would be very welcome! Thanks in advance.

  • $\begingroup$ What, exactly, do YOU mean by a "closed form" solution? This can be solved in terms of the "Lambert W function". Will that do? $\endgroup$ – user247327 Dec 1 '15 at 13:20
  • $\begingroup$ I mean that I would like to explicitly find $x=f(\alpha,\beta,\kappa,\gamma)$, without requiring any iterative procedure or numerical algorithm. Is that possible with the "Lambert W function"? $\endgroup$ – guiomar Dec 1 '15 at 13:27

Ok, so I checked this "Lambert W function", which turns out to be some kind of composite of functions that provide a solution for equations of the form $xe^x=a$. However, the equation I am trying to solve is actually of the form $e^{-cx}=a\frac{x-t}{x-s}$, with $c=-\kappa$, $a=-\frac{\gamma}{\beta}$, $s=\frac{\alpha}{\beta}$ and $t=0$. A solution to this kind of equation requires a generalization of the Lamber W function, as detailed in the following paper: http://arxiv.org/pdf/1408.3999v2.pdf

Unfortunately, this becomes far too complex for the application I am seeking. Thanks anyway for your help!


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