A concave optimization-based approach for sparse multiobjective programming

Abstract

The paper is concerned with multiobjective sparse optimization problems, i.e. the problem of simultaneously optimizing several objective functions and where one of these functions is the number of the non-zero components (or the $$\ell _0$$-norm) of the solution. We propose to deal with the $$\ell _0$$-norm by means of concave approximations depending on a smoothing parameter. We state some equivalence results between the original nonsmooth problem and the smooth approximated problem. We are thus able to define an algorithm aimed to find sparse solutions and based on the steepest descent framework for smooth multiobjective optimization. The numerical results obtained on a classical application in portfolio selection and comparison with existing codes show the effectiveness of the proposed approach.