Smoothed hybrid <inline-formula><tex-math id="M1">$ l_p $</tex-math></inline-formula>-<inline-formula><tex-math id="M2">$ l_2 $</tex-math></inline-formula> model for sparse optimization

In this paper, we propose a new regularization term by using a hybrid function of the smoothed sub-one $ l_p $ quasi-norm and $ l_2 $ norm. The new regularization term avoids the non-differentiability of the sub-one $ l_p $ quasi-norm. At the same time, the new model constructed by it can be transformed into solving two convex sub-models, thereby obtaining an approximate global optimal solution. The coefficients within this hybrid regularization term are obtained through data learning. We apply this new regularization term to sparse optimization by transforming the original problem into two sub-problems. By deriving the corresponding optimality conditions, we develop a novel iterative shrinkage threshold algorithm, called P2ISTA, to solve the smoothed hybrid $ l_p $-$ l_2 $ regularization model. Then the convergence of the algorithm is analyzed. To validate the performance of the proposed algorithm, we conduct computational experiments for well-known image deblurring problems and compare with the existing methods including Soft, Half, PQA, and PLAFPAp. Our computational results clearly demonstrate that the proposed P2ISTA is superior in terms of sparse solution and computational effectiveness.