  In binary classification, kernel-free linear or quadratic support vector
machines are proposed to avoid dealing with difficulties such as finding
appropriate kernel functions or tuning their hyper-parameters. Furthermore,
Universum data points, which do not belong to any class, can be exploited to
embed prior knowledge into the corresponding models so that the generalization
performance is improved. In this paper, we design novel kernel-free Universum
quadratic surface support vector machine models. Further, we propose the L1
norm regularized version that is beneficial for detecting potential sparsity
patterns in the Hessian of the quadratic surface and reducing to the standard
linear models if the data points are (almost) linearly separable. The proposed
models are convex such that standard numerical solvers can be utilized for
solving them. Nonetheless, we formulate a least squares version of the L1 norm
regularized model and next, design an effective tailored algorithm that only
requires solving one linear system. Several theoretical properties of these
models are then reported/proved as well. We finally conduct numerical
experiments on both artificial and public benchmark data sets to demonstrate
the feasibility and effectiveness of the proposed models.
