# Proximal Operator and Optimality Conditions for Ramp Loss SVM

arXiv: https://arxiv.org/abs/2011.09059 · HTML: https://mlsanity.com/paper/2011.09059/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2020.

## Built on by (most cited first, 2 of 2)

- [Nonlinear Kernel Support Vector Machine with 0-1 Soft Margin Loss](https://mlsanity.com/paper/2203.00399/) (2022)
- [Kernel Support Vector Machine Classifiers with the ℓ0-Norm Hinge Loss](https://mlsanity.com/paper/2306.13991/) (2023)

## Abstract (arXiv)

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