In this paper, we apply two convex optimization methods, named UFO, for fixed-outline floorplanning. Our approach consists of two stages which are a global distribution stage and a local legalization stage. In the first stage, we first transform modules into circles and use a pull-push model to distribute modules among a fixed outline under the wirelength consideration. Because good results can be obtained after the first stage, we do not need to consider wirelegnth in the second stage; thus, we can devote to legalize modules. To keep the good results of the first stage, we propose a procedure to extract the geometric relations of modules from a layout and record them by constraint graphs. Then, a quadratic function as well as non-overlap and boundary constraints are formulated to determine the locations and shapes of modules. We have implemented the two convex functions on Matlab, and experimental results have demonstrated that UFO clearly outperforms the results reported in the literature on the GSRC benchmark.