Monocular depth estimation is a highly challenging problem that is often addressed with deep neural networks. While these use recognition of high level image features to predict reasonably looking depth maps,the result often has poor metric accuracy. Moreover,the standard feed forward architecture does not allow modification of the prediction based on cues other than the image.In this paper we relax the monocular depth estimation task by proposing a network that allows us to complement image features with a set of auxiliary variables. These allow disambiguation when image features are not enough to accurately pinpoint the exact depth map and can be thought of as a low dimensional parameterization of the surfaces that are reasonable monocular predictions. By searching the parameterization we can combine monocular estimation with traditional photoconsistency or geometry based methods to achieve both visually appealing and metrically accurate surface estimations. Since we relax the problem we are able to work with smaller networks than current architectures. In addition we design a self-supervised training scheme,eliminating the need for ground truth image depth-map pairs. Our experimental evaluation shows that our method generates more accurate depth maps and generalizes better than competing state-of-the-art approaches.