Properties of the solution of L2-support vector machine as a function of regularization parameter

The goal of this paper is to study some mathematical properties of so-called L2 Soft Margin Support Vector Machines (L2-SVMs) for data classification. Their dual formulations build a family of quadratic programming problems depending on one regularization parameter. The dependence of the solution on this parameter is examined. Such properties as continuity, differentiability, monotony, convexity and structure of the solution are investigated. It is shown that the solution and the objective value of the Hard Margin SVM allow estimating the slack variables of the L2-SVMs. Most results deal with the dual problem, but some statements about the primal problem are also formulated (e.g., the behavior and differentiability of slack variables). An ancillary dual problem is used as investigation tool. It is shown that it is in reality a dual formulation of a quasi identical L2-SVM primal.