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Support vector machines with ramp loss (dubbed as $L_r$-SVM) have attractedwide attention due to the boundedness of ramp loss. However, the correspondingoptimization problem is non-convex and the given Karush-Kuhn-Tucker (KKT)conditions are only the necessary conditions. To enrich the optimality theoryof $L_r$-SVM and go deep into its statistical nature, we first introduce andanalyze the proximal operator for ramp loss, and then establish a strongeroptimality conditions: P-stationarity, which is proved to be the first-ordernecessary and sufficient conditions for local minimizer of $L_r$-SVM. Finally,we define the $L_r$ support vectors based on the concept of P-stationary point,and show that all $L_r$ support vectors fall into the support hyperplanes,which possesses the same feature as the one of hard margin SVM.

Proximal Operator and Optimality Conditions for Ramp Loss SVM