Does a logistic regression maximizing likelihood necessarily also maximize AUC over linear models? Given a data set with binary outcomes $y\in\{0,1\}^n$ and some predictor matrix $X\in\mathbb{R}^{n\times p}$, the standard logistic regression model estimates coefficients $\beta_{MLE}$ which maximize the binomial likelihood. When $X$ is full rank $\beta_{MLE}$ is unique; when perfect separation is not present, it is finite.
Does this maximum likelihood model also maximize the ROC AUC (aka $c$-statistic), or does there exist some coefficient estimate $\beta_{AUC} \neq \beta_{MLE}$ which will obtain a higher ROC AUC? If it is true that the MLE does not necessarily maximize ROC AUC, then another way to look at this question is "Is there an alternative to likelihood maximization which will always maximize ROC AUC of a logistic regression?"
I am assuming that models are otherwise the same: we're not adding or removing predictors in $X$, or otherwise changing the model specification, and I'm assuming that the likelihood-maximizing and AUC-maximizing models are using the same link function. 
 A: It is not the case that $\beta_{MLE} = \beta_{AUC}$.
To illustrate this, consider that AUC can written as 
$P(\hat y_1 > \hat y_0 | y_1 = 1, y_0 = 0)$
In otherwords, the ordering of the predictions is the only thing that affects AUC. This is not the case with the likelihood function. So as a mental exercise, suppose we had a single predictors and in our dataset, we don't see perfect separation (i.e., $\beta_{MLE}$ is finite). Now, if we simply take the value of the largest predictor and increase it by some small amount, we will change the likelihood of this solution, but it will not change the AUC, as the ordering should remain the same. Thus, if the old MLE maximized AUC, it will still maximize AUC after changing the predictor, but will no longer maximize the likelihood. 
Thus, at the very least, it is not the case that $\beta_{AUC}$ is not unique; any $\beta$ that preserves the ordering of the estimates achieves the exact same AUC. In general, since the AUC is sensitive to different aspects of the data, I would believe that we should be able to find a case where $\beta_{MLE}$ does not maximize $\beta_{AUC}$. In fact, I'd venture a guess that this happens with high probability. 
EDIT (moving comment into answer)
The next step is to prove that the MLE doesn't necessarily maximize the AUC (which isn't proven yet). One can do this by taking something like predictors 1, 2, 3, 4, 5, 6, $x$ (with $x > 6$) with outcomes 0, 0, 0, 1, 1, 1, 0.  Any positive value of $\beta$ will maximize the AUC (regardless of the value of $x$), but we can chose an $x$ large enough that the $\beta_{MLE} < 0$. 
