Application of smoothing techniques for linear programming twin support vector machines

In this paper, a new unconstrained minimization problem formulation is proposed for linear programming twin support vector machine (TWSVM) classifiers. The proposed formulation leads to two smaller-sized unconstrained minimization problems having their objective functions piecewise differentiable. However, since their objective functions contain the non-smooth “plus” function, two new smoothing approaches are assumed to solve the proposed formulation, and then apply Newton-Armijo algorithm. The idea of our formulation is to reformulate TWSVM as a strongly convex problem by incorporated regularization techniques and then derive smooth 1-norm linear programming formulation for TWSVM to improve robustness. One significant advantage of our proposed algorithm over TWSVM is that the structural risk minimization principle is implemented in the primal problems which embodies the marrow of statistical learning theory. In addition, the solution of two modified unconstrained minimization problems reduces to solving just two systems of linear equations as opposed to solving two quadratic programming problems in TWSVM and TBSVM, which leads to extremely simple and fast algorithm. Our approach has the advantage that a pair of matrix equation of order equals to the number of input examples is solved at each iteration of the algorithm. The algorithm converges from any starting point that can be easily implemented in MATLAB without using any optimization packages. The performance of our proposed method is verified experimentally on several benchmark and synthetic datasets. Experimental results show the effectiveness of our methods in both training time and classification accuracy.