Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control, system identification and machine learning. In this paper, the non-Lipschitz ℓp (0 < p < 1) regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called p-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover, some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed p-thresholding fixed point continuation (p-FPC) algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.
ℓp regularized matrix minimization matrix completion problem p-thresholding operator global optimality condition fixed point continuation algorithm
90C06 90C26 90C46 65F22 65F30
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11401124 and 71271021), the Scientific Research Projects for the Introduced Talents of Guizhou University (Grant No. 201343) and the Key Program of Natural Science Foundation of China (Grant No. 11431002). The authors are thankful to the two anonymous referees for their valuable suggestions and comments that helped us to revise the paper into the present form.
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