Nonlinear extension of multiobjective multiclass Support Vector Machine

Support Vector Machines (SVM) is originally designed for binary classification having high generalization ability, and thus, many kinds of extended models have been investigated for multiclass classification. In this paper, we focus on a multiobjective multiclass SVM which is one of all-together methods, which maximizing all of the geometric margins simultaneously. Although the model is reported to have high generalization ability, it is formulated as a piece-wise linear model. Hence, the model can be applied to only some kinds of classification problems. Therefore, in this paper, we extend the piece-wise linear MMSVM into a nonlinear one to which kernel method can be applied, where weight vectors of the discriminant function are represented by linear sums of the training patterns in the feature space. Moreover, in order to solve the proposed nonlinear model, we introduce a single-objective optimization problem by exploiting the ε-constraint method and adding the limitation of its constraints. The introduced single-objective model can be regarded as a second-order cone programming (SOCP) problem, and its optimal solution is Pareto optimal for the proposed nonlinear MMSVM. Furthermore, through numerical experiments we verify that the nonlinear MMSVM maximizes the geometric margins in the sense of multiobjective optimization, and has good generalization ability.