Linear discrimination using second order conic programming

We propose a new p-norm linear discrimination model that generalizes the model of Bennett and Mangasarian (1992) and reduces to linear programming problem with p-order conic constraints. We demonstrate that the developed model possesses excellent methodological and computational properties (e.g., it does not allow for a null separating hyperplane when the sets are linearly separable, etc). The proposed approach for handling linear programming problems with p-order conic constraints is based on reformulation of p-conic problems as second-order conic programming (SOCP) problems when p is rational. Since such reformulations typically lead to SOCP problems with large numbers of second-order cones, a greedy algorithm based on graph coloring is presented for reducing the number of second-order cones in the corresponding reformulations. A case study on several popular data sets that illustrates the developed model is conducted.