Multiobjective Multiclass Soft-Margin Support Vector Machine Maximizing Pair-Wise Interclass Margins

Abstract

The all together model is one of the support vector machine (SVM) for multiclass classification by using a piece-wise linear function. As a novel all together model, we already proposed a hard-margin multiobjective SVM model for piecewise linearly separable data, which maximizes all of the geometric margins simultaneously for the generalization ability. In addition, we derived a single-objective convex problem whose optimal solution is weakly Pareto optimal for the proposed SVM. However, in the real-world classification problem the data are often piecewise linearly inseparable. Therefore, in this paper we extend the hard-margin SVM for the data by introducing penalty functions for the margin slack variables based on the geometric distances between outliers and the support hyperplanes, and incorporating those functions into the objective functions. Moreover, we derive a single-objective second-order cone programming (SOCP) problem, and show that its optimal solution is weakly Pareto optimal for the proposed soft-margin SVM. Furthermore through numerical experiments we verify that the SOCP model maximizes the geometric margins in the sense of multiobjective optimization.