Efficient Projected Gradient Methods for L 0 Constrained Optimization

Sparse optimization has attracted increasing attention in numerous areas such as compressed sensing, financial optimization and image processing. In this paper, we first consider a special class of l0 constrained optimization problems, which involves box constraints and a singly linear constraint. An efficient approach is provided for calculating the projection over the feasibility set after a careful analysis on the projection subproblem. Then we present several types of projected gradient methods for a general class of l0 constrained optimization problems. Global convergence of the methods are established under suitable assumptions. Finally, we illustrate some applications of the proposed methods for signal recovery and index tracking. Especially for index tracking, we propose a new model subject to an adaptive upper bound on the sparse portfolio weights. The computational results demonstrate that the proposed projected gradient methods are efficient in terms of solution quality.