An augmented Lagrangian approach for cardinality constrained minimization applied to variable selection problems

To solve convex constrained minimization problems, that also include a cardinality constraint, we propose an augmented Lagrangian scheme combined with alternating projection ideas. Optimization problems that involve a cardinality constraint are NP-hard mathematical programs and typically very hard to solve approximately. Our approach takes advantage of a recently developed and analyzed continuous formulation that relaxes the cardinality constraint. Based on that formulation, we solve a sequence of smooth convex constrained minimization problems, for which we use projected gradient-type methods. In our setting, the convex constraint region can be written as the intersection of a finite collection of convex sets that are easy and inexpensive to project. We apply our approach to a variety of over and under determined constrained linear least-squares problems, with both synthetic and real data that arise in variable selection, and demonstrate its effectiveness.