  We investigate a class of constrained sparse regression problem with
cardinality penalty, where the feasible set is defined by box constraint, and
the loss function is convex, but not necessarily smooth. First, we put forward
a smoothing fast iterative hard thresholding (SFIHT) algorithm for solving such
optimization problems, which combines smoothing approximations, extrapolation
techniques and iterative hard thresholding methods. The extrapolation
coefficients can be chosen to satisfy $\sup_k \beta_k=1$ in the proposed
algorithm. We discuss the convergence behavior of the algorithm with different
extrapolation coefficients, and give sufficient conditions to ensure that any
accumulation point of the iterates is a local minimizer of the original
cardinality penalized problem. In particular, for a class of fixed
extrapolation coefficients, we discuss several different update rules of the
smoothing parameter and obtain the convergence rate of $O(\ln k/k)$ on the loss
and objective function values. Second, we consider the case in which the loss
function is Lipschitz continuously differentiable, and develop a fast iterative
hard thresholding (FIHT) algorithm to solve it. We prove that the iterates of
FIHT converge to a local minimizer of the problem that satisfies a desirable
lower bound property. Moreover, we show that the convergence rate of loss and
objective function values are $o(k^{-2})$. Finally, some numerical examples are
presented to illustrate the theoretical results.
