This study investigates the convex optimization frameworks for accurate and fast negative obstacle detection as well as depth estimations that are vital for real-time, superior autonomous vehicle operations. A convex framework is proposed in conjunction with a unique stereo perception approach to manage the complex nature of such invisible obstacles. The proposed stereo configuration incorporates an alignment that displaces the cameras along a vertical baseline to extract useful information pertaining to negative obstacle features for various illustrative terrain settings. The convex framework exploits these properties to evaluate depth jumps in the disparity space image and perform geometrical analysis of potential occlusion regions. The corresponding convex formulations are based on linear matrix inequalities to detect sudden disparity, angle profile, and intensity variations for potential negative obstacles and to estimate the internal depth of the detected obstacles. Experimental results demonstrate that consecutive processes can be structured in a convex framework to efficiently identify negative obstacle attributes at a range of distances for texture varying environments, providing a vital extension to real-time vehicular system implementations with superior detection rates.