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Sparse optimization has seen its advances in recent decades. For scenarioswhere the true sparsity is unknown, regularization turns out to be a promisingsolution. Two popular non-convex regularizations are the so-called $L_0$ normand $L_q$ norm with $q\in(0,1)$, giving rise to extensive research on theirinduced optimization. However, the majority of these work centered around themain function that is twice continuously differentiable and the bestconvergence rate for an algorithm solving the optimization with $q\in(0,1)$ issuperlinear. This paper explores the $L_q$ norm regularized optimization in aunified way for any $q\in[0,1)$, where the main function has a semismoothgradient. In particular, we establish the first-order and the second-orderoptimality conditions under mild assumptions and then integrate the proximaloperator and semismooth Newton method to develop a proximal semismooth Newtonpursuit algorithm. Under the second sufficient condition, the whole sequencegenerated by the algorithm converges to a unique local minimizer. Moreover, theconvergence is superlinear and quadratic if the gradient of the main functionis semismooth and strongly semismooth at the local minimizer, respectively.Hence, this paper accomplishes the quadratic rate for an algorithm designed tosolve the $L_q$ norm regularization problem for any $q\in(0,1)$. Finally, somenumerical experiments have showcased its nice performance when compared withseveral existing solvers.

Revisiting $L_q(0\leq q<1)$ Norm Regularized Optimization