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We investigate a class of constrained sparse regression problem withcardinality penalty, where the feasible set is defined by box constraint, andthe loss function is convex, but not necessarily smooth. First, we put forwarda smoothing fast iterative hard thresholding (SFIHT) algorithm for solving suchoptimization problems, which combines smoothing approximations, extrapolationtechniques and iterative hard thresholding methods. The extrapolationcoefficients can be chosen to satisfy $\sup_k \beta_k=1$ in the proposedalgorithm. We discuss the convergence behavior of the algorithm with differentextrapolation coefficients, and give sufficient conditions to ensure that anyaccumulation point of the iterates is a local minimizer of the originalcardinality penalized problem. In particular, for a class of fixedextrapolation coefficients, we discuss several different update rules of thesmoothing parameter and obtain the convergence rate of $O(\ln k/k)$ on the lossand objective function values. Second, we consider the case in which the lossfunction is Lipschitz continuously differentiable, and develop a fast iterativehard thresholding (FIHT) algorithm to solve it. We prove that the iterates ofFIHT converge to a local minimizer of the problem that satisfies a desirablelower bound property. Moreover, we show that the convergence rate of loss andobjective function values are $o(k^{-2})$. Finally, some numerical examples arepresented to illustrate the theoretical results.

Smoothing fast iterative hard thresholding algorithm for $\ell_0$ regularized nonsmooth convex regression problem