An unconstrained optimization model is established for assessing roundness errors by the minimum circumscribed circle method based on radial deviation measurement. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the two-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function in order to get the wanted roundness errors by the minimun circumscribed circle assessment. One example is given to verify the theoretical results presented.