In this paper, we propose a novel kernel-free quadratic surface support vector machine with the 0–1 loss function, termed \(L_{0/1}\) -QSSVM, for binary classification. Unlike kernel-based methods, our method explicitly learns a quadratic decision hypersurface in the original feature space, eliminating the need to select kernel functions and their corresponding parameters and making the model more interpretable. Although the 0–1 loss is non-differentiable, we establish first-order optimality conditions via its proximal operator and design an efficient algorithm, called \(L_{0/1}^{*}\) -ADMM, which incorporates a working set strategy inspired by support vectors. We prove that all support vectors lie on the support hypersurfaces, providing a clear geometric display. To further validate the effectiveness of our method, we discuss the convergence and computational complexity of the algorithm, along with the interpretability of the method. Moreover, numerical experiments on artificial and benchmark datasets are conducted, demonstrating that our \(L_{0/1}\) -QSSVM achieves higher accuracy, fewer support vectors, and less computational time compared to state-of-the-art methods.
