Sparse optimization via vector k-norm and DC programming with an application to feature selection for support vector machines

Sparse optimization is about finding minimizers of functions characterized by a number of nonzero components as small as possible, such paradigm being of great practical relevance in Machine Learning, particularly in classification approaches based on support vector machines. By exploiting some properties of the k-norm of a vector, namely, of the sum of its k largest absolute-value components, we formulate a sparse optimization problem as a mixed-integer nonlinear program, whose continuous relaxation is equivalent to the unconstrained minimization of a difference-of-convex function. The approach is applied to Feature Selection in the support vector machine framework, and tested on a set of benchmark instances. Numerical comparisons against both the standard ℓ1 -based support vector machine and a simple version of the Slope method are presented, that demonstrate the effectiveness of our approach in achieving high sparsity level of the solutions without impairing test-correctness.