Training Lagrangian twin support vector regression via unconstrained convex minimization

In this paper, a new unconstrained convex minimization problem formulation is proposed as the Lagrangian dual of the 2-norm twin support vector regression (TSVR). The proposed formulation leads to two smaller sized unconstrained minimization problems having their objective functions piece-wise quadratic and differentiable. It is further proposed to apply gradient based iterative method for solving them. However, since their objective functions contain the non-smooth ‘plus’ function, two approaches are taken: (i) either considering their generalized Hessian or introducing a smooth function in place of the ‘plus’ function, and applying Newton–Armijo algorithm; (ii) obtaining their critical points by functional iterative algorithm. Computational results obtained on a number of synthetic and real-world benchmark datasets clearly illustrate the superiority of the proposed unconstrained Lagrangian twin support vector regression formulation as comparable generalization performance is achieved with much faster learning speed in accordance with the classical support vector regression and TSVR.