In this paper, a reduced quadratic surface support vector machine (RQSSVM) classification model is proposed and solved using the augmented Lagrange method. The new model can effectively handle nonlinearly separable data without kernel function selection and parameter tuning due to its quadratic surface segmentation facility. Meanwhile, the maximum margin term is replaced by an \(L_2\) regularization term and the Hessian of the quadratic surface is reduced to a diagonal matrix. This simplification significantly reduces the number of decision variables and improves computational efficiency. The \(L_1\) loss function is used to transform the problem into a convex composite optimization problem. Then the transformed problem is solved by the Augmented Lagrange method and the non-smoothness of the subproblems is handled by the semi-smooth Newton algorithm. Numerical experiments on artificial and public benchmark datasets show that RQSSVM model not only inherits the superior performance of quadratic surface SVM for segmenting nonlinear surfaces, but also significantly improves the segmentation speed and efficiency.
