The estimation of dense depth maps has become a fundamental module in the pipeline of many visual-based navigation and planning systems. The motivation of our work is to achieve fast and accurate in-situ infrastructure modelling from a monocular camera mounted on an autonomous car. Our technical contribution is in the application of a Lagrangian Multipliers based formulation to minimise an energy that combines a nonconvex data term with adaptive pixel-wise regularisation to yield the final local reconstruction. We advocate the use of constrained optimisation for this task. We shall show it is swift, accurate and simple to implement. Specifically we propose an Augmented Lagrangian (AL) method that markedly reduces the number of iterations required for convergence with more than 50% of reduction in all cases compared to the state-of-the-art approach. As a result, part of this significant saving is invested in improving the accuracy of the depth map. We introduce a novel per pixel inverse depth uncertainty estimation that allows us to apply adaptive regularisation on the initial depth map: high informative inverse depth pixels require less regularisation, however its impact on more uncertain regions can be propagated providing significant improvement on textureless regions. To illustrate the benefits of our approach, we ran our experiments on three synthetic datasets with perfect ground truth for textureless scenes. An exhaustive analysis shows that AL can speed up the convergence up to 90% achieving less than 4cm of error. In addition, we demonstrate the application of the proposed approach on a challenging urban outdoor dataset exhibiting a very diverse and heterogeneous structure.