The nu-support vector classification (nu-SVC) algorithm was shown to work well and provide intuitive interpretations, e.g., the parameter nu roughly specifies the fraction of support vectors. Although nu corresponds to a fraction, it cannot take the entire range between 0 and 1 in its original form. This problem was settled by a non-convex extension of nu-SVC and the extended method was experimentally shown to generalize better than original nu-SVC. However, its good generalization performance and convergence properties of the optimization algorithm have not been studied yet. In this paper, we provide new theoretical insights into these issues and propose a novel nu-SVC algorithm that has guaranteed generalization performance and convergence properties