Support vector machines with ramp loss ( $$L_r$$ -SVM) have attracted considerable attention due to the robustness of loss. However, corresponding optimization problem is non-convex, and given Karush–Kuhn–Tucker (KKT) conditions are only first-order necessary conditions. To enrich optimality theory -SVM, we first introduce analyze proximal operator for loss, then establish a stronger condition:...

This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis truncated Hausdorff distances between polyhedra, we derive new existence and uniqueness theorems for trajectories corresponding various classes control functions acting sets. Then establish quantitative stability results, which provide efficient ...