Nonconvex truncated conditional value at risk-based sparse linear regression

Conditional value at risk (CVaR) is a widely recognized risk measure used to manage data uncertainty within risk management. In this paper, we study a class of sparse linear regression models based on truncated CVaR measure and ℓ0-norm regularization. Due to the nonconvexity and nonsmoothness of the objective functions, as well as the NP-hardness of the problem with the ℓ0-norm regularization, we propose an approximation model that employs a tight relaxation of the ℓ0-norm. The solution equivalence between the proposed model and its approximation model is explored. To efficiently solve the approximation model, we develop a semismooth Newton-based proximal majorization-minimization algorithm. Furthermore, the convergence analysis of the proposed algorithm is presented, and the convergence rate for the reduced CVaR-based sparse linear regression model is established. Moreover, extensive numerical experiments conducted on both synthetic and real datasets validate the stability and effectiveness of the proposed algorithm, demonstrating significant improvements in both sparsity and accuracy compared to existing state-of-the-art methods.