| In this paper, we focus on the all together model of the support vector machine (SVM) for multiclass classification, which constructs a piece-wise linear discriminant function. It is formulated as a single-objective optimization problem maximizing the sum of margins between all pairs of classes, which is defined as the distance between two normalized support hyperplanes parallel to the corresponding discriminant hyperplane where any sample is not contained. However, it is not necessarily equal to the geometric margin defined as the minimal distance of patterns in a pair of classes to the corresponding discriminant hyperplanes. Then, we formulate the proposed model as a multiobjective problem which maximizes all of the margins simultaneously. Moreover, we derive two kinds of single-objective second order cone programming (SOCP) problems based on scalarization approaches, Benson's method and ε-constraint method to solve the proposed multiobjective model, and show that the methods can find Pareto optimal solutions of the model. Furthermore, through numerical experiments we verify the generalization ability of discriminant functions obtained by the proposed SOCP problems. |
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