Efficient projected gradient methods for cardinality constrained optimization

Sparse optimization has attracted increasing attention in numerous areas such as compressed sens-ing, financial optimization and image processing. In this paper, we first consider a special class of cardinality constrained optimization problems, which involves box constraints and a singly linear constraint. An effcient approach is provided for calculating the projection over the feasibility set after a careful analysis on the projec- tion subproblem. Then we present several types of projected gradient methods for a general class of cardinality constrained optimization problems. Global convergence of the methods is established under suitable assump- tions. 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 portfo-lio weights. The computational results demonstrate that the proposed projected gradient methods are effcient in terms of solution quality.