Have MLE estimators for Generalized Pareto Distribution. Given a known value of $c$, how do I calculate $a$ and $b$ using the provided estimators? I am doing research into the three parameter Generalized Pareto Distribution 
$$
f(x|a,b,c) =  \frac 1 b\left(1+a\left(\frac{x-c}{b}\right)\right)^{\big(-1-\frac 1 a\big)}
$$
for finding VaR and CVaR. $x$ is a vector of returns greater than or equal to $c$. The paper Parameter Estimation for 3-parameter gpd by the principle of maximum entropy by Singh and Guo (paper) provides MLE estimators in equations (45) and (46). Given a known value of $c$, how do I calculate $a$ and $b$ using the provided estimators?
 A: Later edit: I give what seems to be a better solution here.

Note that the paper uses a different parameterization from the form given in the question. As Yves noted in comments, it uses $-a$ in place of your $a$ (both are common parameterizations; the only difficulty may be when it is unclear which parameterization is being used). If you convert answers back to your parameterization you'll have to make the corresponding change.
The paper says:

The MLE estimators can be expressed as:
$\sum_{i=1}^n \frac{(x_i-c)/b}{1-a(x_i-c)/b}=\frac{n}{1-a}\qquad\qquad\qquad$    (45)
$\sum_{i=1}^n \ln[1-a(x_i-c)/b]=-na\qquad\,$    (46)
[...] Clearly the likelihood function is maximum with respect to $c$ when $c=x_1$.

In fact there's some minor issues with the exposition; at the point where they set the derivative of the log-likelihood to zero, they're no longer dealing with $a,b$ and $c$ but with the estimators, $\hat{a},\hat{b}$ and $\hat{c}$. So they should really have:
i. $\hat{c} = x_{(1)}$ (at least it didn't appear that the sample was ordered until at that point they suddenly declare $x_1$ to be smallest; better to be explicit)
ii. Given the ML estimate of $c$, the parameters $a$ and $b$ are then estimated by simultaneously solving these two equations:
$\sum_{i=1}^n \frac{(x_i-\hat{c})/\hat{b}}{1-\hat{a}(x_i-\hat{c})/b}=\frac{n}{1-\hat{a}}\qquad\qquad\qquad$    (45)
$\sum_{i=1}^n \ln[1-\hat{a}(x_i-\hat{c})/\hat{b}]=-n\hat{a}\qquad$    (46)
for $\hat{a}$ and $\hat{b}$.
The idea is that given $\hat{c}$, you find $\hat{a}$ and $\hat{b}$ that make equations (45) and (46) true.
If you were to solve this pair of nonlinear equations simultaneously, generally you'd need some iterative scheme set up* that you can update the estimates numerically until (45) and (46) are very close to true.
*(starting with some reasonable guesses, such as method of moments or quantile-based estimates or by assuming $a=0$ and using the resulting ML estimate from an exponential for $\hat{b}$)
It's certainly possible to do so... however, most people would back up a step; rather than taking the derivative and setting it equal to zero and looking for an iterative scheme to solve the equations for $\hat{a}$ and $\hat{b}$, we can simply employ optimization methods to minimize the negative of the log-likelihood function and take as our parameter estimates the values of the parameters that give that minimum. That's what's usually done for the generalized Pareto.
This would again start with some reasonable guesses for  $\hat{a}$ and $\hat{b}$ and iterate to reduce $-\ell=-\log\mathcal{L}$ until some minimum was effectively reached. One benefit of doing so is that once you've found your minimum, it's easier to get second derivative estimates out and so get asymptotic standard errors for the estimates.
[In practice with ML, since the likelihood function might not always be unimodal, it's often a good idea to evaluate it over a grid of plausible values to identify whether there are multiple local minima in $-\ell$ or other issues that might be relevant.]
