Soft data reasoning systems are important components of more general hard and soft data fusion frameworks that should satisfy at a minimum the following four requirements: (1) uncertainty management, (2) semantic richness for soft data representations, (3) computational speed, and (4) robustness against conflicting evidence. Uncertain logic has been recently introduced as a framework for soft data reasoning that can potentially address each of these requirements. Uncertain logic is a First-Order Logic (FOL) reasoning system based on Dempster-Shafer (DS) theoretical models, a mathematical environment that is well suited for addressing requirement (1). The FOL environment directly targets suitable soft data representations, as needed for (2). Uncertain logic, however, still needs improvements for tackling (3) and (4). In this paper, we introduce a method for addressing this need, which is based on formulating an uncertain logic problem as a convex optimization problem. The application of this method is shown through a case study on estimation and tracking in a combined soft and hard data scenario.