Support vector machines with ramp loss (\(L_r\)-SVM) have attracted considerable attention due to the robustness of the ramp loss. However, the corresponding optimization problem is non-convex, and the given Karush–Kuhn–Tucker (KKT) conditions are only first-order necessary conditions. To enrich the optimality theory of \(L_r\)-SVM, we first introduce and analyze the proximal operator for the ramp loss, and then establish a stronger optimality condition: P-stationarity, which is proved to be the first-order necessary and sufficient conditions for the local minimizer of \(L_r\)-SVM. Finally, we define the P-support vectors based on the P-stationary point and show that under mild conditions, all of the P-support vectors for \(L_r\)-SVM are on the two support hyperplanes.

由于斜坡损失的鲁棒性，具有斜坡损失的支持向量机（\（L_r\）- SVM）引起了相当多的关注。然而，相应的优化问题是非凸的，给定的 Karush-Kuhn-Tucker (KKT) 条件只是一阶必要条件。为了丰富\(L_r\)- SVM的最优性理论，我们首先引入并分析了斜坡损失的近端算子，然后建立了一个更强的最优性条件：P-平稳性，它被证明是一阶必要和\(L_r\) -SVM的局部极小值的充分条件。最后，我们定义了基于 P 平稳点的 P 支持向量，并表明在温和条件下，\(L_r\) 的所有 P 支持向量-SVM 位于两个支持超平面上。