  Sparse optimization has seen its advances in recent decades. For scenarios
where the true sparsity is unknown, regularization turns out to be a promising
solution. Two popular non-convex regularizations are the so-called $L_0$ norm
and $L_q$ norm with $q\in(0,1)$, giving rise to extensive research on their
induced optimization. However, the majority of these work centered around the
main function that is twice continuously differentiable and the best
convergence rate for an algorithm solving the optimization with $q\in(0,1)$ is
superlinear. This paper explores the $L_q$ norm regularized optimization in a
unified way for any $q\in[0,1)$, where the main function has a semismooth
gradient. In particular, we establish the first-order and the second-order
optimality conditions under mild assumptions and then integrate the proximal
operator and semismooth Newton method to develop a proximal semismooth Newton
pursuit algorithm. Under the second sufficient condition, the whole sequence
generated by the algorithm converges to a unique local minimizer. Moreover, the
convergence is superlinear and quadratic if the gradient of the main function
is semismooth and strongly semismooth at the local minimizer, respectively.
Hence, this paper accomplishes the quadratic rate for an algorithm designed to
solve the $L_q$ norm regularization problem for any $q\in(0,1)$. Finally, some
numerical experiments have showcased its nice performance when compared with
several existing solvers.
